# On the projections of the multifractal Hewitt-Stromberg dimension functions

Bilel Selmi

###### Abstract

The aim of this paper is to study the behavior of the multifractal Hewitt-
Stromberg dimension functions under projections in Euclidean space. As an
application, we study the multifractal analysis of the projections of a
measure. In particular, we obtain general results for the multifractal
analysis of the orthogonal projections on $m$-dimensional linear subspaces of
a measure $\mu$ satisfying the multifractal formalism which is based on the
Hewitt-Stromberg measures.

MSC-2010: 28A20, 28A78, 28A80

Keyword: Hewitt-Stromberg measures; Modified box-counting dimensions;
Hausdorff dimension; Packing dimension; Projection; Multifractal analysis.

##  1 Introduction and statement of the results

Recently, the projection behavior of dimensions and multifractal spectra of
sets and measures have generated large interest in the mathematical literature
[6, 10, 20, 21, 29, 30, 31, 43, 44]. The multifractal analysis is always
interesting therefore studies are developing: especially the case of
projections of measures, as we did not know till now an exact link between the
dimension of the original object (measure, set,…) and its projections: How to
relate the dimension of the measure with its projections? It is for example
still not known by projecting a multifractal measure on a multifractal set,
such as Cantor’s types, how to deduce the dimension of the projection measure
from the properties of the original one and the dimensions of the projection
and the set we project on it. Dimensional properties of projections of sets
and measures have been investigated for decades. The first significant work in
this area was the result of Marstrand [34], to which the Hausdorff dimension
of a planar set is preserved under typical orthogonal projections. This result
was later generalized to higher dimensions by Kaufman [33], Mattila [35], and
Hu and Taylor [26]. They obtained similar results for the Hausdorff dimension
of a measure. It is natural to seek projection results for other notions of
dimension. However, examples show that the direct analogue of Marstrand’s
projection theorem is not valid for box-counting and packing dimensions,
though there are non-trivial lower bounds on the dimensions of the
projections. The fact that the box-counting and packing dimensions of the
projections of a Borel set $E$ are constant for almost all subspaces $V$ was
established by Falconer and Mattila in [18] and Falconer and Howroyd in [15,
16, 17, 28]. Such constant value, given by a dimension profile of $E$, was
specified somewhat indirectly. For packing dimensions, this is given in terms
of the suprema of dimension profiles of measures supported by $E$ which in
turn are given by critical parameters for certain almost sure pointwise limits
[16]. The approach in [17, 28] defines box-counting dimension profiles in
terms of weighted packings subject to constraints. However, despite these
substantial advances for fractal sets, only very little is known about the
multifractal structure of projections of measures, except a paper by O’Neil
[41]. Later, in [5] Barral and Bhouri, studied the multifractal analysis of
the orthogonal projections on $m$-dimensional linear subspaces of singular
measures on $\mathbb{R}^{n}$ satisfying the classical multifractal formalism.
The result of O’Neil was later generalized by Selmi et al. in [10, 11, 12, 46,
47].

The notion of singularity exponents or spectrum and generalized dimensions are
the major components of the multifractal analysis. They were introduced with a
view of characterizing the geometry of measure and to be linked with the
multifractal spectrum which is the map which affects the Hausdorff or packing
dimension of the iso-Hölder sets

$E_{\mu}(\alpha)=\left\\{x\in\operatorname{supp}\mu\;\Big{|}\;\lim_{r\to
0}\frac{\log\big{(}\mu B(x,r)\big{)}}{\log r}=\alpha\right\\}$

for a given $\alpha\geq 0$ and $\operatorname{supp}\mu$ is the topological
support of probability measure $\mu$ on $\mathbb{R}^{n}$, $B(x,r)$ is the
closed ball of center $x$ and radius $r$. It unifies the multifractal spectra
to the multifractal packing function $b_{\mu}(q)$ and the multifractal packing
function $B_{\mu}(q)$ via the Legendre transform [1, 3, 8, 38], i.e.,

$\dim_{H}\Big{(}E_{\mu}(\alpha)\Big{)}=\inf_{q\in\mathbb{R}}\Big{\\{}q\alpha+b_{\mu}(q)\Big{\\}}=:b_{\mu}^{*}(\alpha)$

and

$\dim_{P}\Big{(}E_{\mu}(\alpha)\Big{)}=\inf_{q\in\mathbb{R}}\Big{\\{}q\alpha+B_{\mu}(q)\Big{\\}}=:B_{\mu}^{*}(\alpha).$

There has been a great interest in understanding the fractal dimensions of
projections of the iso-Hölder sets and measures. The authors in [41, 48]
compared the multifractal packing function $b_{\mu}(q)$ and the multifractal
packing function $B_{\mu}(q)$ of a set $E$ of $\mathbb{R}^{n}$ with respect to
a measure $\mu$ with those of their projections onto $m$-dimensional
subspaces. And they obtained general results for the multifractal analysis of
the orthogonal projections on $m$-dimensional linear subspaces of a measure
$\mu$ satisfying the classical multifractal formalism which is based on the
Hausdorff and packing measures on $\gamma_{n,m}$-almost every linear
$m$-dimensional subspaces for $q>1$, where $\gamma_{n,m}$ is the uniform
measure on $G_{n,m}$, the set of linear $m$-dimensional subspaces of
$\mathbb{R}^{n}$ endowed with its natural structure of a compact metric space,
see [36] for precise definitions of this.

Hewitt-Stromberg measures were introduced in [25, Exercise (10.51)]. Since
then, they have been investigated by several authors, highlighting their
importance in the study of local properties of fractals and products of
fractals. One can cite, for example [23, 24, 32, 40, 50]. In particular,
Edgar’s textbook [13, pp. 32-36] provides an excellent and systematic
introduction to these measures. Such measures appear also explicitly, for
example, in Pesin’s monograph [42, 5.3] and implicitly in Mattila’s text [36].
Motivated by the above papers, the authors in [2, 3] introduced and studied a
multifractal formalism based on the Hewitt-Stromberg measures. However, we
point out that this formalism is completely parallel to Olsen’s multifractal
formalism introduced in [38, 39] which is based on the Hausdorff and packing
measures.

In the present paper we pursue those kinds of studies and consider the
multifractal formalism developed in [41]. We will start by introducing the
multifractal Hewitt-Stromberg measures and dimensions which slightly differ
from those introduced in [2, 3]. Our approach is to consider the behavior of
multifractal Hewitt-Stromberg functions dimensions under projections. These
dimension functions provide a natural mathematical framework within which one
can study multifractals since, for a wide class of measures, they are related
to multifractal spectra via the Legendre transform. As an application, we
study the multifractal analysis based on Hewitt-Stromberg measures of the
projections of a measure and obtain general results for the multifractal
analysis of the orthogonal projections on $m$-dimensional linear subspaces of
a measure $\mu$ satisfying the multifractal formalism.

We will now give a brief description of the organization of the paper. In the
next section, we recall the definitions of the various multifractal dimensions
and measures investigated in the paper. Section 2.1 recalls the multifractal
formalism introduced in [41]. In Section 2.2 we introduce the multifractal
Hewitt-Stromberg measures and separator functions which slightly differ from
those introduced in [2, 3], and study their properties. Section 3 contain our
main results. The proofs are given in Sections 4-5. The paper is concluded
with Section 6 which contains an application of our main results.

##  2 Preliminaries

###  2.1 Multifractal Hausdorff measure and packing measure and dimensions

We start by recalling the multifractal formalism introduced by O’Neil in [41].
The key ideas behind the fine multifractal formalism in [41] are certain
measures of Hausdorff-packing type which are tailored to see only the
multifractal decomposition sets $E_{\mu}(\alpha)$. These measures are natural
multifractal generalizations of the centered Hausdorff measure and the packing
measure and are motivated by the $\tau_{\mu}$-function which appears in the
multifractal formalism. We first recall the definition of the multifractal
Hausdorff measure and the the multifractal packing measure. Let $\mu$ be a
compactly supported probability measure on $\mathbb{R}^{n}$. For
$q,t\in\mathbb{R}$, $E\subseteq\operatorname{supp}\mu$ and $\delta>0$, we
define the multifractal packing pre-measure,

$\overline{{\mathscr{P}}}^{q,t}_{\mu,\delta}(E)=\displaystyle\sup\left\\{\sum_{i}\mu\left(B\left(x_{i},\frac{r_{i}}{3}\right)\right)^{q}r_{i}^{t}\right\\},$

where the supremum is taken over all $\delta$-packing of $E$,

$\overline{{\mathscr{P}}}^{q,t}_{\mu}(E)=\displaystyle\inf_{\delta>0}\overline{{\mathscr{P}}}^{q,t}_{\mu,\delta}(E).$

The function $\overline{{\mathscr{P}}}^{q,t}_{\mu}$ is increasing but not
$\sigma$-subadditive. That is the reason why O’Neil introduced the
modification of the multifractal packing measure ${\mathscr{P}}^{q,t}_{\mu}$:

${\mathscr{P}}^{q,t}_{\mu}(E)=\inf_{E\subseteq\bigcup_{i}E_{i}}\sum_{i}\overline{\mathscr{P}}^{q,t}_{\mu}(E_{i}).$

In a similar way, we define the Hausdorff measure,

${{\mathscr{H}}}^{{q},t}_{{\mu},\delta}(E)=\displaystyle\inf\left\\{\sum_{i}\mu\Big{(}B(x_{i},3r_{i})\Big{)}^{q}~{}r_{i}^{t}\;\Big{|}\;\Big{(}B(x_{i},r_{i})\Big{)}_{i}\text{is
a $\delta$-covering of}\;E\right\\},$

and

${{\mathscr{H}}}^{{q},t}_{{\mu}}(E)=\displaystyle\sup_{\delta>0}{{\mathscr{H}}}^{q,t}_{{\mu},\delta}(E).$

The functions ${\mathscr{P}}^{q,t}_{\mu}$ and ${\mathscr{H}}^{q,t}_{\mu}$ are
metric outer measures and thus measures on the family of Borel subsets of
$\mathbb{R}^{n}$. An important feature of the pre-packing, packing and
Hausdorff measure is that
${\mathscr{P}}^{q,t}_{\mu}\leq{\overline{\mathscr{P}}}^{q,t}_{\mu}$ and there
is a constant $c$ depending also on the dimension of the ambient space, such
that ${\mathscr{H}}^{q,t}_{\mu}\leq c~{}{\mathscr{P}}^{q,t}_{\mu}$ (see [41]).

The functions $\overline{{\mathscr{P}}}^{q,t}_{\mu}$,
${\mathscr{P}}^{q,t}_{\mu}$ and ${\mathscr{H}}^{q,t}_{\mu}$ assign, in the
usual way, a dimension to each subset $E$ of $\operatorname{supp}\mu$. They
are respectively denoted by $\Lambda_{\mu}^{q}(E)$, $B_{\mu}^{q}(E)$ and
$b_{\mu}^{q}(E)$.

  1. 1.

There exists a unique number $\Lambda_{\mu}^{q}(E)\in[-\infty,+\infty]$ such
that

$\overline{{\mathscr{P}}}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<\Lambda_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&\Lambda_{\mu}^{q}(E)<t.\end{matrix}\right.$

  2. 2.

There exists a unique number $B_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

${\mathscr{P}}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<B_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&B_{\mu}^{q}(E)<t.\end{matrix}\right.$

  3. 3.

There exists a unique number $b_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

${\mathscr{H}}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<b_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&b_{\mu}^{q}(E)<t.\end{matrix}\right.$

The number $b_{\mu}^{q}(E)$ is an obvious multifractal analogue of the
Hausdorff dimension $\dim_{H}(E)$ of $E$ whereas $B_{\mu}^{q}(E)$ and
$\Lambda_{\mu}^{q}(E)$ are obvious multifractal analogues of the packing
dimension $\dim_{P}(E)$ and the pre-packing dimension $\Delta(E)$ of $E$
respectively. In fact, it follows immediately from the definitions that

$\dim_{H}(E)=b_{\mu}^{0}(E),\;\;\;\dim_{P}(E)=B_{\mu}^{0}(E)\quad\text{and}\quad\Delta(E)=\Lambda_{\mu}^{0}(E).$

We note that for all $q\in\mathbb{R}$

$b_{\mu}^{q}(\emptyset)=B_{\mu}^{q}(\emptyset)=\Lambda_{\mu}^{q}(\emptyset)=-\infty,$

and if $\mu(E)=0$, then

$b_{\mu}^{q}(E)=B_{\mu}^{q}(E)=\Lambda_{\mu}^{q}(E)=-\infty\quad\text{for}\quad
q>0.$

Next, we define the separator functions $\Lambda_{\mu}$, $B_{\mu}$ and
$b_{\mu}$ : $\mathbb{R}\rightarrow[-\infty,+\infty]$ by,

$\Lambda_{\mu}(q)=\Lambda_{\mu}^{q}(\operatorname{supp}\mu)$,
$B_{\mu}(q)=B_{\mu}^{q}(\operatorname{supp}\mu)$ and
$b_{\mu}(q)=b_{\mu}^{q}(\operatorname{supp}\mu).$

It is well known that the functions $\Lambda_{\mu}$, $B_{\mu}$ and $b_{\mu}$
are decreasing. The functions $\Lambda_{\mu}$, $B_{\mu}$ convex and satisfying
$b_{\mu}\leq B_{\mu}\leq\Lambda_{\mu}.$

###### Proposition 2.1

[41] Let $\mu$ be compactly supported probability measure on $\mathbb{R}^{n}$.
Then, we have

  1. 1.

For $q<1$, $0\leq b_{\mu}(q)\leq B_{\mu}(q)\leq\Lambda_{\mu}(q).$

  2. 2.

$b_{\mu}(1)=B_{\mu}(1)=\Lambda_{\mu}(1)=0.$

  3. 3.

For $q>1$, $b_{\mu}(q)\leq B_{\mu}(q)\leq\Lambda_{\mu}(q)\leq 0.$

###### Remark 2.1

The multifractal Hausdorff and packing measures introduced by O’Neil are
different from those developed by Olsen [38], although, when $\mu$ satisfies a
doubling condition, the multifractal measures are equivalent.

The reader is referred to as Olsen’s classical text [38] (see also [41, 48])
for an excellent and systematic discussion of the multifractal Hausdorff and
packing measures and dimensions.

###  2.2 Multifractal Hewitt-Stromberg measures and separator functions

Our main reason for modifying the definitions of Attia et al. in [2, 3] is to
allow us to prove results for non necessary doubling measures. One main cause
and motivation is the fact that such characteristics is not in fact preserved
under projections. In the following, we will set up, for $q,t\in\mathbb{R}$
and a compactly supported probability measure $\mu$ on $\mathbb{R}^{n}$, the
lower and upper multifractal Hewitt-Stromberg measures
${\mathsf{H}}_{\mu}^{q,t}$ and ${\mathsf{P}}_{\mu}^{q,t}$. For
$E\subseteq\operatorname{supp}\mu$, the pre-measure of $E$ is defined by

${\mathsf{C}}_{\mu}^{q,t}(E)=\limsup_{r\to 0}M_{\mu,r}^{q}(E)~{}r^{t},$

where

$M_{\mu,r}^{q}(E)=\sup\left\\{\displaystyle\sum_{i}\mu\left(B\left(x_{i},\frac{r}{3}\right)\right)^{q}\;\Big{|}\;\Big{(}B(x_{i},r)\Big{)}_{i}\;\text{is
a packing of}\;E\right\\}.$

Observe that ${\mathsf{C}}_{\mu}^{q,t}$ is increasing and
${\mathsf{C}}_{\mu}^{q,t}(\emptyset)=0$. However it is not $\sigma$-additive.
For this, we introduce the ${\mathsf{P}}_{\mu}^{q,t}$-measure defined by

${\mathsf{P}}_{\mu}^{q,t}(E)=\inf\left\\{\displaystyle\sum_{i}{\mathsf{C}}_{\mu}^{q,t}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\text{and
the}\;E_{i}^{\prime}\text{s are bounded}\right\\}.$

In a similar way we define

${\mathsf{L}}_{\mu}^{q,t}(E)=\liminf_{r\to 0}M_{\mu,r}^{q}(E)~{}r^{t}.$

Since ${\mathsf{L}}_{\mu}^{q,t}$ is not countably subadditive, one needs a
standard modification to get an outer measure. Hence, we modify the definition
as follows

${\mathsf{H}}_{\mu}^{q,t}(E)=\inf\left\\{\displaystyle\sum_{i}{\mathsf{L}}_{\mu}^{q,t}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\text{and
the}\;E_{i}^{\prime}\text{s are bounded}\right\\}$

The measure $\mathsf{H}^{q,t}_{\mu}$ is of course a multifractal
generalization of the lower $t$-dimensional Hewitt-Stromberg measure
${\mathsf{H}}^{t}$, whereas $\mathsf{P}^{q,t}_{\mu}$ is a multifractal
generalization of the upper $t$-dimensional Hewitt-Stromberg measures
${\mathsf{P}}^{t}$. In fact, it is easily seen that, for $t>0$, one has

$\mathsf{H}^{0,t}_{\mu}={\mathsf{H}}^{t}\quad\text{and}\quad\mathsf{P}^{0,t}_{\mu}={\mathsf{P}}^{t}.$

The following result describes some of the basic properties of the
multifractal Hewitt-Stromberg measures including the fact that
${\mathsf{H}}_{\mu}^{q,t}$ and ${\mathsf{P}}_{\mu}^{q,t}$ are Borel metric
outer measures and summarises the basic inequalities satisfied by the
multifractal Hewitt-Stromberg measures, the multifractal Hausdorff measure and
the multifractal packing measure.

###### Theorem 2.1

Let $q,t\in\mathbb{R}$. Then for every set $E\subseteq\mathbb{R}^{n}$ we have

  1. 1.

the set functions $\mathsf{H}_{\mu}^{q,t}$ and $\mathsf{P}_{\mu}^{q,t}$ are
metric outer measures and thus they are measures on the Borel algebra.

  2. 2.

There exists an integer $\xi\in\mathbb{N}$, such that

${\mathscr{H}}^{q,t}_{\mu}(E)\leq\xi\mathsf{H}_{\mu}^{q,t}(E)$

and

$\displaystyle\mathsf{L}_{\mu}^{q,t}(E)\leq\mathsf{C}_{\mu}^{q,t}(E)\leq\overline{{\mathscr{P}}}^{q,t}_{\mu}(E)$
$\displaystyle\;\;\;\mathrm{\vee{l}}\quad\quad\quad\;\mathrm{\vee
l}\quad\quad\quad\;\mathrm{\vee l}$
$\displaystyle\mathsf{H}_{\mu}^{q,t}(E)\leq\mathsf{P}_{\mu}^{q,t}(E)\leq{\mathscr{P}}^{q,t}_{\mu}(E).$

The proof of the first part is straightforward and mimics that in [2, Theorem
2.1]. The proof of second part is a straightforward application of
Besicovitch’s covering theorem and we omit it here (we can see also [2,
Theorem 2.1]).

The measures $\mathsf{H}^{q,t}_{\mu}$ and $\mathsf{P}^{q,t}_{\mu}$ and the
pre-measures $\mathsf{L}^{q,t}_{\mu}$ and ${\mathsf{C}}_{\mu}^{q,t}$ assign in
the usual way a multifractal dimension to each subset $E$ of $\mathbb{R}^{n}$,
they are respectively denoted by $\mathsf{b}_{\mu}^{q}(E)$,
$\mathsf{B}_{\mu}^{q}(E)$, ${\mathscr{L}}_{\mu}^{q}(E)$ and
$\mathsf{\Delta}_{\mu}^{q}(E)$.

###### Proposition 2.2

Let $q\in\mathbb{R}$ and $E\subseteq\mathbb{R}^{n}$. Then

  1. 1.

there exists a unique number ${\mathsf{b}}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

$\mathsf{H}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{b}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{b}}_{\mu}^{q}(E)<t,\end{matrix}\right.$

  2. 2.

there exists a unique number ${\mathsf{B}}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

$\mathsf{P}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{B}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{B}}_{\mu}^{q}(E)<t,\end{matrix}\right.$

  3. 3.

there exists a unique number
${\mathsf{\Delta}}_{\mu}^{q}(E)\in[-\infty,+\infty]$ such that

$\mathsf{C}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathsf{\Delta}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathsf{\Delta}}_{\mu}^{q}(E)<t,\end{matrix}\right.$

  4. 4.

there exists a unique number $\mathscr{L}_{\mu}^{q}(E)\in[-\infty,+\infty]$
such that

${\mathsf{L}}^{q,t}_{\mu}(E)=\left\\{\begin{matrix}\infty&\text{if}&t<{\mathscr{L}}_{\mu}^{q}(E),\\\
\\\ 0&\text{if}&{\mathscr{L}}_{\mu}^{q}(E)<t.\end{matrix}\right.$

In addition, we have

${\mathsf{b}}_{\mu}^{q}(E)\leq{\mathsf{B}}_{\mu}^{q}(E)\leq{\mathsf{\Delta}}_{\mu}^{q}(E).$

The number ${\mathsf{b}}_{\mu}^{q}(E)$ is an obvious multifractal analogue of
the lower Hewitt-Stromberg dimension $\underline{\dim}_{MB}(E)$ of $E$ whereas
${\mathsf{B}}_{\mu}^{q}(E)$ is an obvious multifractal analogues of the upper
Hewitt-Stromberg dimension $\overline{\dim}_{MB}(E)$ of $E$. Observe that the
number ${\mathscr{L}}_{\mu}^{q}(E)$ is an obvious multifractal analogue of the
lower box-dimension $\underline{\dim}_{B}(E)$ of $E$ whereas
${\mathsf{\Delta}}_{\mu}^{q}(E)$ is an obvious multifractal analogues of the
upper box-dimension $\overline{\dim}_{B}(E)$ of $E$. In fact, it follows
immediately from the definitions that

${\mathscr{L}}_{\mu}^{0}(E)=\underline{\dim}_{B}(E),\;\;{\mathsf{\Delta}}_{\mu}^{0}(E)=\overline{\dim}_{B}(E)$

and

${\mathsf{b}}_{\mu}^{0}(E)=\underline{\dim}_{MB}(E),\;\;{\mathsf{B}}_{\mu}^{0}(E)=\overline{\dim}_{MB}(E).$

See [3, 32, 40] for precise definitions of these dimensions.

###### Remark 2.2

It follows from Theorem 2.1 that

$\displaystyle{\mathscr{L}}_{\mu}^{q}(E)\leq{\mathsf{\Delta}}_{\mu}^{q}(E)\leq\Lambda_{\mu}^{q}(E)$
$\displaystyle\;\;\mathrm{\vee l}\;\qquad\quad\mathrm{\vee
l}\;\qquad\quad\mathrm{\vee l}$ $\displaystyle b_{\mu}^{q}(E)\leq$
$\displaystyle\;{\mathsf{b}}_{\mu}^{q}(E)\leq{\mathsf{B}}_{\mu}^{q}(E)\leq
B_{\mu}^{q}(E).$

The definition of these dimension functions makes it clear that they are
counterparts of the $\tau_{\mu}$-function which appears in the multifractal
formalism. This being the case, it is important that they have the properties
described by the physicists. The next theorem shows that these functions do
indeed have some of these properties.

###### Theorem 2.2

Let $q\in\mathbb{R}$ and $E\subseteq\mathbb{R}^{n}$.

  1. 1.

The functions $q\mapsto\mathsf{H}^{q,t}_{\mu}(E)$,
$\mathsf{P}^{q,t}_{\mu}(E)$, $\mathsf{C}^{q,t}_{\mu}(E)$ are decreasing.

  2. 2.

The functions $t\mapsto\mathsf{H}^{q,t}_{\mu}(E)$,
$\mathsf{P}^{q,t}_{\mu}(E)$, $\mathsf{C}^{q,t}_{\mu}(E)$ are decreasing.

  3. 3.

The functions $q\mapsto{{\mathsf{b}}}^{q}_{\mu}(E)$,
${{\mathsf{B}}}^{q}_{\mu}(E)$, ${{\mathsf{\Delta}}}^{q}_{\mu}(E)$ are
decreasing.

  4. 4.

The functions $q\mapsto{{\mathsf{B}}}^{q}_{\mu}(E)$,
${{\mathsf{\Delta}}}^{q}_{\mu}(E)$ are convex.

The proof of this is straightforward and mimics that in [3, Theorem 3].

We note that for all $q\in\mathbb{R}$

${\mathsf{b}}_{\mu}^{q}(\emptyset)={\mathsf{B}}_{\mu}^{q}(\emptyset)={\mathsf{\Delta}}_{\mu}^{q}(\emptyset)=-\infty,$

and if $\mu(E)=0$, then

${\mathsf{b}}_{\mu}^{q}(E)={\mathsf{B}}_{\mu}^{q}(E)={\mathsf{\Delta}}_{\mu}^{q}(E)=-\infty\quad\text{for}\quad
q>0.$

Next, we define the separator functions ${\mathsf{\Delta}}_{\mu}$,
${\mathsf{B}}_{\mu}$ and ${\mathsf{b}}_{\mu}$ :
$\mathbb{R}\rightarrow[-\infty,+\infty]$ by,

${\mathsf{\Delta}}_{\mu}(q)={\mathsf{\Delta}}_{\mu}^{q}(\operatorname{supp}\mu)$,
${\mathsf{B}}_{\mu}(q)={\mathsf{B}}_{\mu}^{q}(\operatorname{supp}\mu)$ and
${\mathsf{b}}_{\mu}(q)={\mathsf{b}}_{\mu}^{q}(\operatorname{supp}\mu).$

The multifractal formalism based on the measures ${\mathsf{H}}^{q,t}_{\mu}$
and ${\mathsf{P}}^{q,t}_{\mu}$ and the dimension functions $\mathsf{b}_{\mu}$,
$\mathsf{B}_{\mu}$ and $\mathsf{\Delta}_{\mu}$ provides a natural, unifying
and very general multifractal theory which includes all the hitherto
introduced multifractal parameters, i.e., the multifractal spectra functions
$\alpha\mapsto\mathsf{f}_{\mu}(\alpha)=:\underline{\dim}_{MB}E_{\mu}(\alpha)$
and
$\alpha\mapsto\mathsf{F}_{\mu}(\alpha)=:\overline{\dim}_{MB}E_{\mu}(\alpha)$,
the multifractal box dimensions. The dimension functions $\mathsf{b}_{\mu}$
and $\mathsf{B}_{\mu}$ are intimately related to the spectra functions
$\mathsf{f}_{\mu}$ and $\mathsf{F}_{\mu}$ (see [3]), whereas the dimension
function $\mathsf{\Delta}_{\mu}$ is closely related to the upper box spectrum
(more precisely, to the upper multifractal box dimension function
$\overline{\tau}_{\mu}$, see Proposition 5.2).

##  3 Main results

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$. In the following, we require an alternative characterization
of the upper and lower multifractal box-counting dimensions of $\mu$ in terms
of a potential obtained by convolving $\mu$ with a certain kernel. For this
purpose let us introduce some notations. For $1\leq s\leq n$ and $r>0$ we
define

$\begin{array}[]{llll}\phi_{r}^{s}:&\mathbb{R}^{n}&\longrightarrow&\mathbb{R}\\\
&x&\longmapsto&\min\Big{\\{}1,\>r^{s}|x|^{-s}\Big{\\}},\end{array}$

and

$\mu\ast\phi_{r}^{s}(x)=\int\min\Big{\\{}1,\>r^{s}|x-y|^{-s}\Big{\\}}d\mu(y).$

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $1\leq s\leq n$
and $q>1$, write

$N_{\mu,r}^{q,s}(E)=\int_{E}\Big{(}\mu\ast\phi_{r/3}^{s}(x)\Big{)}^{q-1}d\mu(x),$

and

$\overline{\tau}_{\mu}^{q,s}(E)=\displaystyle\limsup_{r\to 0}\frac{\log
N_{\mu,r}^{q,s}(E)}{-\log
r}\quad\text{and}\quad\underline{\tau}_{\mu}^{q,s}(E)=\displaystyle\liminf_{r\to
0}\frac{\log N_{\mu,r}^{q,s}(E)}{-\log r}.$

These definitions are, frankly, messy, indirect and unappealing. In an attempt
to make the concept more attractive, we present here an alternative approach
to the dimensions $\underline{\tau}_{\mu}^{q,s}$ and
$\overline{\tau}_{\mu}^{q,s}$, and their applications to projections in terms
of a potential obtained by convolving $\mu$ with a certain kernel. For $E$ a
compact subset of $\operatorname{supp}\mu$ we can try to decompose $E$ into a
countable number of pieces $E_{1},E_{2},...$ in such a way that the largest
piece has as small a dimension as possible. The present approach was first
used by Falconer in [14, Section 3.3] and further developed by O’Neil and
Selmi in [41, 48]. This idea leads to the following modified dimensions in
terms of the convolutions:

$\displaystyle\underline{\mathfrak{T}}_{\mu}^{q,s}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\underline{\tau}_{\mu}^{q,s}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact }\right\\},$
$\displaystyle\overline{\mathfrak{T}}_{\mu}^{q,s}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\overline{\tau}_{\mu}^{q,s}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact }\right\\}$

and

$\underline{\mathfrak{T}}_{\mu}^{s}(q)=\underline{\mathfrak{T}}_{\mu}^{q,s}(\operatorname{supp}\mu)\quad\text{and}\quad\overline{\mathfrak{T}}_{\mu}^{s}(q)=\overline{\mathfrak{T}}_{\mu}^{q,s}(\operatorname{supp}\mu)\quad\text{for
all}\quad s\geq 1.$

Let $m$ be an integer with $0<m\leq n$ and $G_{n,m}$ the Grassmannian manifold
of all $m$-dimensional linear subspaces of $\mathbb{R}^{n}$. Denote by
$\gamma_{n,m}$ the invariant Haar measure on $G_{n,m}$ such that
$\gamma_{n,m}(G_{n,m})=1$. For $V\in G_{n,m}$, we define the projection map
$\pi_{V}:\mathbb{R}^{n}\longrightarrow V$ as the usual orthogonal projection
onto $V$. Then, the set $\\{\pi_{V},\;V\in G_{n,m}\\}$ is compact in the space
of all linear maps from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$ and the
identification of $V$ with $\pi_{V}$ induces a compact topology for $G_{n,m}$.
Also, for a Borel probability measure $\mu$ with compact support
$supp\mu\subset\mathbb{R}^{n}$ and for $V\in G_{n,m}$, we denote by $\mu_{V},$
the projection of $\mu$ onto $V$, i.e.,

$\mu_{V}(A)=\mu\circ\pi_{V}^{-1}(A)\quad\forall A\subseteq V.$

Since $\mu$ is compactly supported and $supp\mu_{V}=\pi_{V}(supp\mu)$ for all
$V\in G_{n,m}$, then, for any continuous function
$f:V\longrightarrow\mathbb{R}$, we have

$\displaystyle\int_{V}fd\mu_{V}=\int f(\pi_{V}(x))d\mu(x),$

whenever these integrals exist. Then for all $V\in G_{n,m}$,
$x\in\mathbb{R}^{n}$ and $0<r<1$, we have

$\mu\ast\phi_{r}^{m}(x)=\int\mu_{V}(B(x_{V},r))dV=\int\min\Big{\\{}1,\>r^{m}|x-y|^{-m}\Big{\\}}d\mu(y).$

In the following we compare the lower and upper multifractal Hewitt-Stromberg
dimensions of a set $E$ of $\mathbb{R}^{n}$ with respect to a measure $\mu$
with those of their projections onto $m$-dimensional subspaces.

###### Theorem 3.1

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$ and
$E\subseteq\operatorname{supp}\mu$. For $q\leq 1$ and all $V\in G_{n,m}$, we
have

${\mathscr{L}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathscr{L}}_{\mu}^{q}(E),\;\;\;{\mathsf{\Delta}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{\Delta}}_{\mu}^{q}(E)$

and

${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{b}}_{\mu}^{q}(E),\;\;\;{\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{B}}_{\mu}^{q}(E).$

The following result presents alternative expressions of the multifractal
dimension functions ${\mathscr{L}}_{\mu}^{q}(E)$ and
${\mathsf{\Delta}}_{\mu}^{q}(E)$ of a set $E$ and that of its orthogonal
projections.

###### Theorem 3.2

Let $E$ be a compact subset of $\operatorname{supp}\mu$. Then, we have

  1. 1.

for all $q>1$ and $V\in G_{n,m},$

${\mathscr{L}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\underline{\tau}_{\mu}^{q,m}(E)\geq{\mathscr{L}}_{\mu}^{q}(E)$

and

${\Delta}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\overline{\tau}_{\mu}^{q,m}(E)\geq{\mathsf{\Delta}}_{\mu}^{q}(E).$

  2. 2.

For all $1<q\leq 2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

${\mathsf{\Delta}}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),{\mathsf{\Delta}}_{\mu}^{q}(E)\Big{)}$

and

${\mathscr{L}}_{\mu_{V}}^{q}(\pi_{V}(E))=\underline{\tau}_{\mu}^{q,m}(E).$

  3. 3.

For all $q>2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

    1.

    2. (a)

If ${\mathsf{\Delta}}_{\mu}^{q}(E)\geq-m$ then
${\mathsf{\Delta}}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)={\mathsf{\Delta}}_{\mu}^{q}(E).$

    3. (b)

${\mathscr{L}}_{\mu_{V}}^{q}(\pi_{V}(E))=\max\Big{(}m(1-q),\underline{\tau}_{\mu}^{q,m}(E)\Big{)}.$

In Theorem 3.3 we show that ${\mathsf{B}}_{\mu}^{q}(E)$ is preserved under
$\gamma_{n,m}$-almost every orthogonal projection for $q>1$.

###### Theorem 3.3

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q>1$.

  1. 1.

For all $V\in G_{n,m}$, we have

${\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq{\mathsf{B}}_{\mu}^{q}(E).$

  2. 2.

If $1<q\leq 2$, one has

${\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\mathfrak{T}}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),{\mathsf{B}}_{\mu}^{q}(E)\Big{)},\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

  3. 3.

If $q>2$ and $({E}_{i})_{i}$ is a cover of $E$ by a countable collection of
compact sets is such that ${\mathsf{\Delta}}_{\mu}^{q}(E_{i})\geq-m$ for all
$i$, then

${\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\mathfrak{T}}_{\mu}^{q,m}(E)={\mathsf{B}}_{\mu}^{q}(E),\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

The next theorem enables us to study the lower multifractal Hewitt-Stromberg
dimension of the projection of sets on $m$-dimensional linear subspaces for
$q>1$. In particular, we prove that ${\mathsf{b}}_{\mu}^{q}(E)$ is not
preserved under $\gamma_{n,m}$-almost every orthogonal projection for $q>1$.

###### Theorem 3.4

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q>1$.

  1. 1.

For all $V\in G_{n,m}$, we have

${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq{\mathsf{b}}_{\mu}^{q}(E).$

  2. 2.

If $1<q\leq 2$, one has

${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))=\underline{\mathfrak{T}}_{\mu}^{q,m}(E),\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

  3. 3.

If $q>2$, then

${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))=\max\Big{(}m(1-q),\underline{\mathfrak{T}}_{\mu}^{q,m}(E)\Big{)},\;\text{for
$\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}.$

For an integer $s$ with $1\leq m\leq s<n$, we define the $s$-energy of a
measure $\mu$ by

$\displaystyle I_{s}(\mu)=\displaystyle\int\int|x-y|^{-s}d\mu(x)d\mu(y).$

Frostman [22] showed that the Hausdorff dimension of a Borel subset $E$ of
$\mathbb{R}^{n}$ is the supremum of the positive reals $s$ for which there
exists a Borel probability measure $\mu$ charging $E$ and for which the
$s$-energy of $\mu$ is finite. This characterization is used by Kaufmann [33]
and Mattila [36] to prove their results on the preservation of the Hausdorff
dimension. The condition $I_{s}(\mu)<\infty$ implies that $\dim_{H}(\mu)\geq
s$. On the other hand, if $\mu(B(x,r))\leq r^{s}$, for all $x$ and all
sufficiently small $r$ then $\mu$ has a finite $s$-energy. Notice that Mattila
[36] proved that if $I_{m}(\mu)$ is finite, then for almost every
$m$-dimensional subspace $V$, the measure $\mu_{V}$ is absolutely continuous
with respect to Lebesgue measure $\mathcal{L}_{V}^{m}$ on $V$ identified with
$\mathbb{R}^{m}$ and $\mu_{V}\in L^{2}(V)$, where
$\mathcal{L}_{V}^{m}(E)=\mathcal{L}^{m}(E\cap V)$ for
$E\subset\mathbb{R}^{m}$. The following result enables us to describe the
behavior of large measures under projection.

###### Theorem 3.5

Suppose that $\mu$ is a compactly supported Radon measure on $\mathbb{R}^{n}$
and $0<m\leq s<n$ are such that $I_{s}(\mu)<\infty.$ Then

  1. 1.

if $2m<s<n$, then $\text{for $\gamma_{n,m}$-almost every}\;\;V\in G_{n,m}$ and
$q\geq 0$

${\mathsf{b}}_{\mu_{V}}({q})={\mathsf{B}}_{\mu_{V}}({q})=m(1-q),$

  2. 2.

if $m\leq s\leq 2m$, then $\text{for $\gamma_{n,m}$-almost every}\;\;V\in
G_{n,m}$ and $q\geq 0$

$m(1-q)\leq{\mathsf{b}}_{\mu_{V}}(q)\leq{\mathsf{B}}_{\mu_{V}}(q)\leq\max\Big{(}m(1-q),-\frac{sq}{2}\Big{)}.$

###### Remark 3.1

Fix $0<m\leq n$ and let $\mu$ be a self-similar measure on $\mathbb{R}^{n}$
with support equal to $K$ such that $\dim_{P}(K)=s\leq m$. Let $q\geq 0$ and
$({E}_{i})_{i}$ be a cover of $E$ by a countable collection of compact sets is
such that ${\mathsf{\Delta}}_{\mu}^{q}(E_{i})\geq-m$ for all $i$. By using
Theorems 3.3 and 3.4 and [41, Corollary 5.12], we have for
$\gamma_{n,m}$-almost every $V\in G_{n,m}$

$\displaystyle
B_{{\mu}_{V}}({q})={\mathsf{b}}_{{\mu}_{V}}({q})={\mathsf{B}}_{{\mu}_{V}}({q})=b_{{\mu}_{V}}({q})=b_{{\mu}}({{q}})=$
$\displaystyle{\mathsf{b}}_{{\mu}}({{q}})={\mathsf{B}}_{{\mu}}({{q}})=B_{{\mu}}({q})=\underline{\mathfrak{T}}_{\mu}^{m}(q)=\overline{\mathfrak{T}}_{\mu}^{m}(q).$

##  4 Proof of Theorem 3.1

When $q<0$ it suffice to observe that if
$\Big{(}B\left(x_{i},r\right)\Big{)}_{i\in\mathbb{N}}$ is a centered packing
of $\pi_{V}(E)$ then $\Big{(}B\left(y_{i},r\right)\Big{)}_{i\in\mathbb{N}}$ is
a centered packing of $E$ where $y_{i}\in E$ is such that
$x_{i}=\pi_{V}\left(y_{i}\right)$ which implies that
$\mu_{V}(B\left(x_{i}\frac{r}{3}\right))^{q}\leq\mu(B\left(y_{i},\frac{r}{3}\right))^{q}$.
This easily gives the desired result.

For $0\leq q\leq 1,$ fix $V\in G_{n,m}$ and let
$\Big{(}B(x_{i},r)\Big{)}_{i\in\mathbb{N}}$ be a packing of $\pi_{V}(E)$. For
each $i$ we consider the collection of balls as follows

$\left\\{B\left(y,\frac{r}{9}\right)\;\Big{|}\;y\in
E\cap\pi_{V}^{-1}\left(V\cap B\left(x_{i},\frac{r}{3}\right)\right)\right\\},$

and let $E_{i}=E\cap\pi_{V}^{-1}\big{(}V\cap B(x_{i},\frac{r}{3})\big{)}$. So
Besicovitch’s covering theorem (see [41, Theorem 2.2]) provides a positive
integer $\xi=\xi(n)$ and index sets, $I_{i1},I_{i2},...,I_{i\xi}$ such that
$E_{i}\subset\bigcup_{j=1}^{\xi}\bigcup_{i\in
I_{j}}B\left(y,\frac{r}{9}\right)$ and the subset
$\Big{\\{}B(y,\frac{r}{9})\;\big{|}\;y\in I_{ij}\Big{\\}}$ is a disjoint
family for each $j$. Also, we observe that for a fixed $i$ and $j$, we may
make a simple volume estimate to further subdivide $J_{ij}$ into at most
$7^{n}$ disjoint subfamilies, $J_{ij1},J_{ij2},...,J_{ij7^{n}}$ such that for
each $j$ and $k$,
$\left\\{B\left(y,\frac{r}{3}\right)\;\big{|}\;y\in\cup_{i}I_{ijk}\right\\}$
is a centered packing of $E$. Since $0<q\leq 1$, we have

$\displaystyle\sum_{i}\mu_{V}\left(B\left(x_{i},\frac{r}{3}\right)\right)^{q}$
$\displaystyle\leq$
$\displaystyle\sum_{i}\mu\left(\bigcup_{j=1}^{\xi}\bigcup_{k=1}^{7^{n}}\bigcup_{y\in
I_{ijk}}B\left(y,\frac{r}{9}\right)\right)^{q}$ $\displaystyle\leq$
$\displaystyle\sum_{i}\sum_{j=1}^{\xi}\sum_{k=1}^{7^{n}}\sum_{y\in
I_{ijk}}\mu\left(B\left(y,\frac{r}{9}\right)\right)^{q}$ $\displaystyle\leq$
$\displaystyle\xi 7^{n}M_{\mu,\frac{r}{3}}^{q}(E).$

This implies that

$M_{\mu_{V},r}^{q}(\pi_{V}(E))\leq\xi 7^{n}M_{\mu,\frac{r}{3}}^{q}(E).$

Letting $r\downarrow 0$, now yields

${\mathsf{L}}_{\mu_{V}}^{q,t}(\pi_{V}(E))\leq 3^{t}\xi
7^{n}{\mathsf{L}}_{\mu}^{q,t}(E)\quad\text{and}\quad{\mathsf{C}}_{\mu_{V}}^{q,t}(\pi_{V}(E))\leq
3^{t}\xi 7^{n}{\mathsf{C}}_{\mu}^{q,t}(E).$

We deduce from the previous inequalities that

${\mathscr{L}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathscr{L}}_{\mu}^{q}(E)\quad\text{and}\quad{\mathsf{\Delta}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{\Delta}}_{\mu}^{q}(E).$

Now, let $t>{\mathsf{B}}_{\mu}^{q}(E)$ which implies that
$\mathsf{P}_{\mu}^{q,t}(E)<\infty$, then we can choose $(E_{i})_{i}$ a
covering of $E$ such that
$\displaystyle\sum_{i}{\mathsf{C}}_{\mu}^{q,t}(E_{i})<1.$ We therefore
conclude that $\pi_{V}(E)\subseteq\displaystyle\bigcup_{i}\pi_{V}(E_{i})$ and

$\displaystyle\mathsf{P}_{\mu_{V}}^{q,t}(\pi_{V}(E))$ $\displaystyle\leq$
$\displaystyle\displaystyle\sum_{i}{\mathsf{C}}_{\mu_{V}}^{q,t}(\pi_{V}(E_{i}))\leq
3^{t}\xi 7^{n}\sum_{i}{\mathsf{C}}_{\mu}^{q,t}(E_{i})\leq 3^{t}\xi
7^{n}<\infty.$

This implies that

${\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq t\quad\text{for all}\quad
t>{\mathsf{B}}_{\mu}^{q}(E).$

We now infer that

$\displaystyle{\mathsf{B}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{B}}_{\mu}^{q}(E).$
(4.1)

The proof of the statement
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq{\mathsf{b}}_{\mu}^{q}(E)$ is
identical to the proof of the statement (4.1) and is therefore omitted.

##  5 Proof of Theorems 3.2, 3.3, 3.4 and 3.5

We present the tools, as well as the intermediate results, which will be used
in the proof of our main results.

###  5.1 Preliminary results

Let $\mu$ be a compactly supported probability measure on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$. Recall that the upper and lower multifractal box-counting
dimensions $\overline{\tau}_{\mu}^{q}$ and $\underline{\tau}_{\mu}^{q}$ of $E$
are defined respectively by

$\overline{\tau}_{\mu}^{q}(E)=\displaystyle\limsup_{r\to 0}\frac{\log
M_{\mu,r}^{q}(E)}{-\log
r}\quad\text{and}\quad\underline{\tau}_{\mu}^{q}(E)=\displaystyle\liminf_{r\to
0}\frac{\log M_{\mu,r}^{q}(E)}{-\log r}.$

For technical convenience we shall assume that,, if $r\in(0,1)$: $\frac{\log
0}{-\log r}=-\infty$.

The next result is essentially a restatement of [5, Proposition 4.2] and [9,
Proposition 5.1] (see also [19, Lemma 2.6 (a)] and [49]), and has recently
been obtained in [48, Proposition 4.2].

###### Proposition 5.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$. For $q>1$, we have

$\underline{\tau}_{\mu}^{q}(E)=\liminf_{r\to 0}\frac{1}{-\log
r}\log\int_{E}\mu\left(B\left(x,\frac{r}{3}\right)\right)^{q-1}d\mu(x)$

and

$\overline{\tau}_{\mu}^{q}(E)=\limsup_{r\to 0}\frac{1}{-\log
r}\log\int_{E}\mu\left(B\left(x,\frac{r}{3}\right)\right)^{q-1}d\mu(x).$

In the next we investigate the relation between the lower and upper
multifractal Hewitt-Stromberg functions $\mathsf{b}_{\mu}$ and
$\mathsf{B}_{\mu}$ and the multifractal box dimension, the multifractal
packing dimension and the multifractal pre-packing dimension.

###### Proposition 5.2

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}$. Then for every $E\subseteq\operatorname{supp}\mu$
we have

${\mathscr{L}}_{\mu}^{q}(E)=\underline{\tau}_{\mu}^{q}(E)\qquad\text{and
}\qquad{\mathsf{\Delta}}_{\mu}^{q}(E)=\overline{\tau}_{\mu}^{q}(E)=\Lambda_{\mu}^{q}(E).$

Proof. We will prove the first equality, the second one is similar. Suppose
that

$\underline{\tau}_{\mu}^{q}(E)>{\mathscr{L}}_{\mu}^{q}(E)+\epsilon\quad\text{
for some}\quad\epsilon>0.$

Then we can find $\delta>0$ such that for any $r\leq\delta$,

$M_{\mu,r}^{q}(E)~{}r^{{\mathscr{L}}_{\mu}^{q}(E)+\epsilon}>1\quad\text{and
then}\quad\mathsf{L}_{\mu}^{q,{\mathscr{L}}_{\mu}^{q}(E)+\epsilon}\geq 1$

which is a contradiction. We therefore infer

$\underline{\tau}_{\mu}^{q}(E)\leq{\mathscr{L}}_{\mu}^{q}(E)+\epsilon\;\;\text{
for any}\;\;\epsilon>0.$

The proof of the following statement

$\underline{\tau}_{\mu}^{q}(E)\geq{\mathscr{L}}_{\mu}^{q}(E)-\epsilon\;\;\text{
for any}\;\;\epsilon>0$

is identical to the proof of the above statement and is therefore omitted.

We have the following additional property.

###### Proposition 5.3

Let $q\in\mathbb{R}$ and $\mu$ be a compact supported Borel probability
measure on $\mathbb{R}^{n}$. Then for every $E\subseteq\operatorname{supp}\mu$
we have

${\mathsf{b}}_{\mu}^{q}(E)=\inf\left\\{\sup_{i}{\mathscr{L}}_{\mu}^{q}(E_{i})\;\Big{|}\;\displaystyle
E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are bounded in
}\;\;\mathbb{R}^{n}\right\\}$

and

${\mathsf{B}}_{\mu}^{q}(E)=\inf\left\\{\sup_{i}{\mathsf{\Delta}}_{\mu}^{q}(E_{i})\;\Big{|}\;\displaystyle
E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are bounded in
}\;\;\mathbb{R}^{n}\right\\}.$

Proof. Denote

$\beta=\inf\left\\{\sup_{i}{\mathscr{L}}_{\mu}^{q}(E_{i})\;\Big{|}\;\displaystyle
E\subseteq\bigcup_{i}E_{i},\;\;E_{i}\;\;\text{are bounded in
}\;\;\mathbb{R}^{n}\right\\}.$

Assume that $\beta<{\mathsf{b}}_{\mu}^{q}(E)$ and take
$\alpha\in(\beta,{\mathsf{b}}_{\mu}^{q}(E))$. Then we can choose $\\{E_{i}\\}$
of bounded subset of $E$ such that $E\subseteq\cup_{i}E_{i}$, and
$\sup_{i}{\mathscr{L}}_{\mu}^{q}(E_{i})<\alpha$. Observe that
$\mathsf{L}_{\mu}^{q,\alpha}(E_{i})=0$ which implies that
${{\mathsf{H}}}_{\mu}^{q,\alpha}(E)=0$. It is a contradiction. Now suppose
that ${\mathsf{b}}_{\mu}^{q}(E)<\beta$, then, for any
$\alpha\in({\mathsf{b}}_{\mu}^{q}(E),\beta),$ we have
${\mathsf{H}}_{\mu}^{q,\alpha}(E)=0$. Thus, there exists $\\{E_{i}\\}$ of
bounded subset of $E$ such that $E\subseteq\cup_{i}E_{i}$, and
$\sup_{i}{\mathsf{L}}_{\mu}^{q,\alpha}(E_{i})<\infty$. We conclude that,
$\sup_{i}{\mathscr{L}}_{\mu}^{q}(E_{i})\leq\alpha$. It is also a
contradiction. The proof of the second statement is identical to the proof of
the statement in the first part and is therefore omitted.

The following proposition is a consequence of Propositions 5.2 and 5.3.

###### Proposition 5.4

Let $E$ be a compact subset of $\operatorname{supp}\mu$ and $q\in\mathbb{R}$.
One has

$\displaystyle\mathsf{b}_{\mu}^{q}(E)=\inf\left\\{\sup_{1\leq
i<\infty}\underline{\tau}_{\mu}^{q}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact}\;\right\\}$

and

$\displaystyle\mathsf{B}_{\mu}^{q}(E)$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\overline{\tau}_{\mu}^{q}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact}\;\right\\}$ $\displaystyle=$
$\displaystyle\inf\left\\{\sup_{1\leq
i<\infty}\Lambda_{\mu}^{q}(E_{i})\;\Big{|}\;E\subseteq\bigcup_{i}E_{i}\;\;\text{with
each $E_{i}$ compact}\;\right\\}.$

###### Proposition 5.5

Let $E$ be a subset of $\operatorname{supp}\mu$ and $q\in\mathbb{R}$. Then we
have

$B_{\mu}^{q}(E)=\mathsf{B}_{\mu}^{q}(E).$

Proof. It follows immediately from Proposition 5.4 and [48, Proposition 4.1].

The following result presents alternative expressions of the upper and lower
multifractal box-counting dimensions in terms of the convolutions as well as
general relations between the upper and lower multifractal box-counting
dimensions of a measure and that of its orthogonal projections. This result
has recently been obtained in [48, Theorem 4.1].

###### Theorem 5.1

Let $E$ be a compact subset of $\operatorname{supp}\mu$. Then, we have

  1. 1.

for all $q>1$ and $V\in G_{n,m},$

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\underline{\tau}_{\mu}^{q,m}(E)\quad\text{and}\quad\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\overline{\tau}_{\mu}^{q,m}(E).$

  2. 2.

For all $1<q\leq 2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

$\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)=\max\Big{(}m(1-q),\overline{\tau}_{\mu}^{q}(E)\Big{)}$

and

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\underline{\tau}_{\mu}^{q,m}(E).$

  3. 3.

For all $q>2$ and $\gamma_{n,m}$-almost every $V\in G_{n,m},$

    1.

    2. (a)

If $-m\leq\overline{\tau}_{\mu}^{q}(E)$ then
$\overline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\overline{\tau}_{\mu}^{q,m}(E)=\overline{\tau}_{\mu}^{q}(E).$

    3. (b)

$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E))=\max\Big{(}m(1-q),\underline{\tau}_{\mu}^{q,m}(E)\Big{)}.$

The assertion (2) is essentially a restatement of the main result of Hunt et
al. in [27] and Falconer et al. in [19, Theorem 3.9]. The assertion (3)
extends the result of Hunt and Kaloshin (of Falconer and O’Neil) to the case
$q>2$ untreated in their work.

###  5.2  Proof of Theorem 3.2

Follows directly from Proposition 5.2 and Theorem 5.1.

###  5.3  Proof of Theorem 3.3

The proof of the first part of Theorem 3.3 follows immediately from Theorem
5.1 (1.), Proposition 5.3 and since
$\overline{\tau}_{\mu}^{q,m}(E)\geq\overline{\tau}_{\mu}^{q}(E)$. By using
Proposition 5.5, then the proof of the second and the third part of Theorem
3.3 has recently been obtained in [48, Theorem 3.2].

###  5.4  Proof of Theorem 3.4

  1. 1.

The proof of the first part of Theorem 3.4 follows immediately from Theorem
5.1 (1.), Proposition 5.3 and since
$\underline{\tau}_{\mu}^{q,m}(E)\geq\underline{\tau}_{\mu}^{q}(E)$.

  2. 2.

If $s>\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$ we may cover $E$ by a countable
collection of sets $E_{i}$, which we may take to be compact, such that
$\underline{\tau}_{\mu}^{q,m}(E_{i})<s$. By using Theorem 5.1 (2.), we have
$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\leq s$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$ Proposition 5.4 implies that
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq s$ for $\gamma_{n,m}$-almost every
$V\in G_{n,m}$ and so,
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$
for $\gamma_{n,m}$-almost every $V\in G_{n,m}.$

Now, if $s<\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$. Fix $V\in G_{n,m}$ and
let $(\widetilde{E}_{i})_{i}$ be a cover of the compact set $\pi_{V}(E)$ by a
countable collection of compact sets. Put for each $i$,
$E_{i}=E\cap\pi_{V}^{-1}(\widetilde{E}_{i}),$ then
$\sup_{i}\underline{\tau}_{\mu}^{q,m}(E_{i})>s$. By using Theorem 5.1 (1.), we
have $\sup_{i}\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\geq s$ and
$\sup_{i}\underline{\tau}_{\mu_{V}}^{q}(\widetilde{E}_{i})\geq s$, this
implies that ${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq s$. Therefore, we
obtain
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$.

  3. 3.

First we suppose that $\underline{\mathfrak{T}}_{\mu}^{q,m}(E)\geq m(1-q)$. If
$s>\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$ we may cover $E$ by a countable
collection of sets $E_{i}$, which we may take to be compact, such that
$\underline{\tau}_{\mu}^{q,m}(E_{i})<s$. By using Theorem 5.1 (3.) we have
$\underline{\tau}_{\mu_{V}}^{q}(\pi_{V}(E_{i}))\leq s$ for
$\gamma_{n,m}$-almost every $V\in G_{n,m}.$ Proposition 5.4 implies that
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq s$ for $\gamma_{n,m}$-almost every
$V\in G_{n,m}$ and so,
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\leq\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$
for $\gamma_{n,m}$-almost every $V\in G_{n,m}.$ The proof of
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))\geq\underline{\mathfrak{T}}_{\mu}^{q,m}(E)$
for all $V\in G_{n,m},$ is identical to the proof of the above statement and
is therefore omitted.

Now, we suppose that $\underline{\mathfrak{T}}_{\mu}^{q,m}(E)\leq m(1-q)$. It
follows from Proposition 5.4 and Theorem 5.1 (3.) that
${\mathsf{b}}_{\mu_{V}}^{q}(\pi_{V}(E))=m(1-q)$ for $\gamma_{n,m}$-almost
every $V\in G_{n,m}.$

###  5.5 Proof of Theorem 3.5

The proof of Theorem 3.5 is straightforward from Remark 2.2 and [41, Corollary
4.4].

##  6 Application

When $\mu$ obeys the multifractal formalism over some interval, we are
interested in knowing whether or not this property is preserved after
orthogonal projections on $\gamma_{n,m}$-almost every linear $m$-dimensional
subspaces. In this section we study the behavior of projections of measures
obeying to the multifractal formalism which is based on the Hewitt-Stromberg
measures. More specifically, we prove that for $q>1$ if the multifractal
formalism holds for $\mu$ at $\alpha=-{\mathsf{B}}^{\prime}_{\mu}(q)$, it
holds for $\mu_{V}$ for $\gamma_{n,m}$-almost every $V\in G_{n,m}$. The
multifractal analysis is a natural framework to describe geometrically the
heterogeneity in the distribution at small scales of positive and finite
compactly supported Borel measures on $\mathbb{R}^{n}$. Specifically, for such
a measure $\mu$, this heterogeneity can be classified by considering the iso-
Hölder sets

$\mathscr{X}_{\mu}(\alpha)=\left\\{x\in\operatorname{supp}\mu\;\Big{|}\;\lim_{r\to
0}\frac{\log\Big{(}\mu B(x,3r)\Big{)}}{\log r}=\alpha\right\\}$

which form a partition of $\operatorname{supp}\mu$. Then the singularity
spectrum of the measure $\mu$ is the mapping
$\alpha\mapsto\dim_{(.)}\mathscr{X}_{\mu}(\alpha)$ and this spectrum provides
a geometric hierarchy between the sets $\mathscr{X}_{\mu}(\alpha)$. We mention
that in the last decade there has been a great interest for the multifractal
analysis and positive results have been written in various situations, see for
example [1, 3, 7, 8, 38, 39, 41, 48]. The function ${\mathsf{B}}_{\mu}(q)$ is
related to the multifractal spectrum of the measure $\mu$. More precisely,
$f^{*}(\alpha)=\inf_{\beta}\big{(}\alpha\beta+f(\beta)\big{)}$ denotes the
Legendre transform of the function $f,$ it has been proved in [3] a lower and
upper bound estimate of the singularity spectrum using the Legendre transform
of the function ${\mathsf{B}}_{\mu}(q)$. In this section, we will work with
the following formalism which a consequence of the multifractal formalism
developed in [2, 3].

###### Theorem 6.1

Let $\mu$ be a compactly supported Borel probability measure on
$\mathbb{R}^{n}$ and $q\in\mathbb{R}$. Suppose that

  1. 1.

there exists a nontrivial (Frostman) measure $\nu_{q}$ satisfying

$\nu_{q}(B(x,r))\leq\mu(B(x,3r))^{q}~{}r^{{\mathsf{B}}_{\mu}(q)}\quad\text{where}\quad
x\in\operatorname{supp}\mu,\;\;0<r<1.$

  2. 2.

${\mathsf{B}}_{\mu}$ is differentiable at $q$.

Then, for any $\alpha=-{\mathsf{B}}^{\prime}_{\mu}(q)$

$\displaystyle\dim_{H}\mathscr{X}_{\mu}\big{(}\alpha\big{)}$ $\displaystyle=$
$\displaystyle\underline{\dim}_{MB}\mathscr{X}_{\mu}\big{(}\alpha\big{)}=\overline{\dim}_{MB}\mathscr{X}_{\mu}\big{(}\alpha\big{)}=\dim_{P}\mathscr{X}_{\mu}\big{(}\alpha\big{)}$
$\displaystyle=$ $\displaystyle
B^{*}_{\mu}\big{(}\alpha\big{)}=b^{*}_{\mu}\big{(}\alpha\big{)}={\mathsf{b}}^{*}_{\mu}\big{(}\alpha\big{)}={\mathsf{B}}^{*}_{\mu}\big{(}\alpha\big{)}.$

The following proposition has recently been obtained in [41, Lemma 3.2].

###### Proposition 6.1

Let $\mu$ be a compactly supported Borel probability measure on
$\mathbb{R}^{n}$. For $q\geq 1$ and all $V\in G_{n,m}$, we have

$b_{\mu_{V}}(q)\geq\max\Big{(}m(1-q),b_{\mu}(q)\Big{)}.$

In the following, we study the validity of the multifractal formalism under
projection. More specifically, we obtain general result for the multifractal
analysis of the orthogonal projections on $m$-dimensional linear subspaces of
measure $\mu$ satisfying the multifractal formalism which is based on the
Hewitt-Stromberg measures.

###### Theorem 6.2

Let $\mu$ be a compactly supported Borel probability measure on
$\mathbb{R}^{n}$ and $q>1$. Suppose that

$(\mathsf{H}_{1})$ there exists a nontrivial (Frostman) measure $\nu_{q}$
satisfying

$\nu_{q}(B(x,r))\leq\mu(B(x,3r))^{q}~{}r^{{\mathsf{B}}_{\mu}(q)}\quad\text{where}\quad
x\in\operatorname{supp}\mu,\;\;0<r<1,$

$(\mathsf{H}_{2})$ ${\mathsf{B}}_{\mu}$ is differentiable at $q$,

$(\mathsf{H}_{3})$ $({E}_{i})_{i}$ be a cover of $\operatorname{supp}\mu$ by a
countable collection of compact sets is such that
$b_{\mu}^{q}(E_{i}\cap\operatorname{supp}\mu)\geq\max(-m,m(1-q))$ for all $i$.

Then, for any $\alpha=-{\mathsf{B}}^{\prime}_{\mu}(q)$ and
$\gamma_{n,m}$-almost every $V\in G_{n,m}$,

$\displaystyle\underline{\dim}_{MB}\mathscr{X}_{\mu}\big{(}\alpha\big{)}=\overline{\dim}_{MB}\mathscr{X}_{\mu_{V}}(\alpha)=\underline{\dim}_{MB}\mathscr{X}_{\mu_{V}}\big{(}\alpha\big{)}=\overline{\dim}_{MB}\mathscr{X}_{\mu}(\alpha)=$
$\displaystyle\dim_{H}\mathscr{X}_{\mu}\big{(}\alpha\big{)}=\dim_{P}\mathscr{X}_{\mu_{V}}\big{(}\alpha\big{)}=\dim_{H}\mathscr{X}_{\mu_{V}}\big{(}\alpha\big{)}=\dim_{P}\mathscr{X}_{\mu}\big{(}\alpha\big{)}=$
$\displaystyle
B^{*}_{\mu}\big{(}\alpha\big{)}=b^{*}_{\mu}\big{(}\alpha\big{)}={\mathsf{b}}^{*}_{\mu}\big{(}\alpha\big{)}={\mathsf{B}}^{*}_{\mu}\big{(}\alpha\big{)}.$

Proof. It follows from Theorems 3.3 and 3.4, Propositions 6.1 and 5.5,
$(\mathsf{H}_{1})$ and $(\mathsf{H}_{3})$ that, for $\gamma_{n,m}$-almost
every $V\in G_{n,m}$,

$B_{{\mu}_{V}}({q})={\mathsf{b}}_{{\mu}_{V}}({q})={\mathsf{B}}_{{\mu}_{V}}({q})=b_{{\mu}_{V}}({q})=b_{{\mu}}({{q}})={\mathsf{b}}_{{\mu}}({{q}})={\mathsf{B}}_{{\mu}}({{q}})=B_{{\mu}}({q}).$
(6.1)

The hypothesis $(\mathsf{H}_{1})$, (6.1) and the proof of Lemma 3.2 in [41]
ensure that, there exists a positive constant $c$ such that

$0<\mathscr{H}_{\mu}^{q,{\mathsf{B}}_{\mu}(q)}(\operatorname{supp}\mu)\leq
c~{}\mathscr{H}_{\mu_{V}}^{q,{\mathsf{B}}_{\mu_{V}}(q)}(\operatorname{supp}\mu_{V}),$
for $\gamma_{n,m}$-almost every $V\in G_{n,m}$.

By using [41, Theorem 5.1] there exists a Frostman measure $\nu_{q}$
satisfying

$\nu_{q}(B(y,r))\leq\mu_{V}(B(y,3r))^{q}~{}r^{{\mathsf{B}}_{\mu_{V}}(q)}\quad\text{where}\quad
y\in\operatorname{supp}\mu_{V},\;\;0<r<1.$

Now, it follows from the hypothesis $(\mathsf{H}_{2})$ and Theorem 6.1 that

$\displaystyle\underline{\dim}_{MB}\mathscr{X}_{\mu_{V}}(\alpha)\geq\dim_{H}\mathscr{X}_{\mu_{V}}\big{(}\alpha\big{)}\geq
q\alpha+{\mathsf{B}}_{\mu}(q),$ (6.2)

$\text{for}\;\gamma_{n,m}\text{-almost every}\;V\in G_{n,m}$. Clearly the
assumption (6.1) give that

$\displaystyle\overline{\dim}_{MB}\mathscr{X}_{\mu_{V}}(\alpha)=\dim_{P}\mathscr{X}_{\mu_{V}}\big{(}\alpha\big{)}$
$\displaystyle\leq$
$\displaystyle{\mathsf{B}}^{*}_{\mu_{V}}\big{(}\alpha\big{)}={\mathsf{B}}^{*}_{\mu}\big{(}\alpha\big{)},$
(6.3)

for $\gamma_{n,m}$-almost every $V\in G_{n,m}$. Now combining (6.2), (6.3) and
the equalities (6.1) we get the desired result.

## References

  * [1] N. Attia and B. Selmi. Relative multifractal box-dimensions. Filomat, (2019), vol. 33, pp. 2841-2859.
  * [2] N. Attia and B. Selmi. Regularities of multifractal Hewitt-Stromberg measures. Commun. Korean Math. Soc., (2019), vol. 34, pp. 213-230.
  * [3] N. Attia and B. Selmi. A multifractal formalism for Hewitt-Stromberg measures, Journal of Geometric Analysis, (to appear). https://doi.org/10.1007/s12220-019-00302-3
  * [4] F. Bahroun and I. Bhouri. _Multifractals and projections_. Extracta Math., (2006), vol. 21, pp. 83-91.
  * [5] J. Barral and I. Bhouri. _Multifractal analysis for projections of Gibbs and related measures_. Ergodic Theory Dynam. Systems., (2011), vol. 31, pp. 673-701.
  * [6] J. Barral and D.J. Feng. _Projections of planar Mandelbrot random measures_. Adv. Math., (2018), vol. 325, pp. 640-718.
  * [7] J. Barral, F. Ben Nasr and J. Peyrière. Comparing multifractal formalism: the neighbouring box condition. Asian J. Math., (2003), vol. 7, pp. 149-166.
  * [8] F. Ben Nasr, I. Bhouri and Y. Heurteaux. The validity of the multifractal formalism: results and examples. Adv. Math., (2002), vol. 165, pp. 264-284.
  * [9] I. Bhouri. _On the projections of generalized upper $L^{q}$-spectrum_. Chaos, Solitons and Fractals, (2009), vol. 42, pp. 1451-1462.
  * [10] Z. Douzi and B. Selmi. _Multifractal variation for projections of measures_. Chaos, Solitons and Fractals, (2016), vol. 91, pp. 414-420.
  * [11] Z. Douzi and B. Selmi. _On the projections of mutual multifractal spectra_. arXiv:1805.06866v1, (2018).
  * [12] Z. Douzi and B. Selmi. _On the projections of the mutual multifractal Rényi dimensions_. Anal. Theory Appl., (to appear).
  * [13] G. A. Edgar. Integral, probability, and fractal measures. Springer-Verlag, New York, (1998).
  * [14] K. J. Falconer. Fractal geometry: mathematical foundations and applications. Chichester (1990). Wiley.
  * [15] K. J. Falconer. A capacity approach to box and packing dimensions of projections of sets and exceptional directions. J. Fractal Geom., (to appear).
  * [16] K. J. Falconer and J. Howroyd. Projection theorems for box and packing dimensions. Math. Proc. Cambridge Philos. Soc., (1996), vol. 119, pp. 287-295.
  * [17] K. J. Falconer and J.D. Howroyd. _Packing Dimensions of Projections and Dimensions Profiles_. Math. Proc. Cambridge Philos. Soc., (1997), vol 121, pp. 269-286.
  * [18] K. J. Falconer and P. Mattila. _The Packing Dimensions of Projections and Sections of Measures_. Math. Proc. Cambridge Philos. Soc., (1996), vol. 119, pp. 695-713.
  * [19] K. J. Falconer and T.C. O’Neil. _Convolutions and the geometry of multifractal measures_. Math. Nachr., (1999), vol. 204, pp. 61-82.
  * [20] K. J. Falconer and P. Mattila. Strong Marstrand theorems and dimensions of sets formed by subsets of hyperplanes. J. Fractal Geom., (2016), vol. 3, pp. 319-329.
  * [21] K. J. Falconer and X. Jim. Exact dimensionality and projections of random self-similar measures and sets. J. Lond. Math. Soc., (2014), vol. 90, pp.388-412.
  * [22] O. Frostman. _Potential d equilibre et capacit des ensembles avec quelques applications à la th orie des fonctions Meddel_. Lunds Univ. Math. Sem., (1935), vol. 3, pp. 1-118.
  * [23] H. Haase. A contribution to measure and dimension of metric spaces. Math. Nachr., (1985), vol. 124, pp. 45-55.
  * [24] H. Haase. Open-invariant measures and the covering number of sets. Math. Nachr., (1987), vol. 134, pp. 295-307.
  * [25] E. Hewitt and K. Stromberg. Real and abstract analysis. A modern treatment of the theory of functions of a real variable. Springer-Verlag, New York, (1965).
  * [26] X. Hu and J. Taylor. _Fractal properties of products and projections of measures in $\mathbb{R}^{n}$_. Math. Proc. Cambridge Philos. Soc., (1994), vol. 115, pp. 527-544.
  * [27] B. R. Hunt and V.Y. Kaloshin. _How projections affect the dimension spectrum of fractal measures_. Nonlinearity, (1997), vol 10, pp. 1031-1046.
  * [28] J. D. Howroyd. Box and packing dimensions of projections and dimension profiles. Math. Proc. Cambridge Philos. Soc., (2001) vol. 130, pp. 135 160.
  * [29] E. Järvenpää and M. Järvenpä. _Linear mappings and generalized upper spectrum for dimensions_. Nonlinearity. (1998), vol. 45, pp. 55-77.
  * [30] E. Järvenpää, M. Järvenpä, F. Ledrappier and M. Leikas. _One-dimensional families of projections_. Nonlinearity, (2008), vol. 21, pp. 453-464.
  * [31] E. Järvenpää, M. Järvenpä and T. Keleti. _Hausdorff Dimension and Non-degenerate Families of Projections_. J. Geom. Anal., (2014), vol. 24, pp. 2020-2034.
  * [32] S. Jurina, N. MacGregor, A. Mitchell, L. Olsen and A. Stylianou. On the Hausdorff and packing measures of typical compact metric spaces. Aequationes Math., (2018), vol. 92, pp. 709-735.
  * [33] R. Kaufman. _On Hausdorff dimension of projections_. Mathematika, (1968), vol. 15, pp. 153-155.
  * [34] J.M. Marstrand. _Some fundamental geometrical properties of plane sets of fractional dimensions_. Proc. Lond. Math. Soc., (1954), vol. 4, pp. 257-302.
  * [35] P. Mattila. _Hausdorff dimension, orthogonal projections and intersections with planes_. Ann. Acad. Sci. Fenn. Math., (1975), vol. 1, pp. 227-244.
  * [36] P. Mattila. _The Geometry of Sets and Measures in Euclidean Spaces_. Cambridge University Press, Cambrdige, (1995).
  * [37] L. Olsen. Multifractal dimensions of product measures. Math. Proc. Camb. Phil. Soc., (1996), vol. 120, pp. 709-734.
  * [38] L. Olsen. A multifractal formalism. Adv. Math., (1995), vol. 166, pp. 82-196.
  * [39] L. Olsen. Dimension Inequalities of Multifractal Hausdorff Measures and Multifractal Packing Measures. Math. Scand., (2000), vol. 86, pp. 109-129.
  * [40] L. Olsen. On average Hewitt-Stromberg measures of typical compact metric spaces. Mathematische Zeitschrift, (2019), vol. 293, pp. 1201 1225
  * [41] T.C. O’Neil. _The multifractal spectra of projected measures in Euclidean spaces_. Chaos, Solitons and Fractals, (2000), vol. 11, pp. 901-921.
  * [42] Y. Pesin, Dimension theory in dynamical systems, Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, (1997).
  * [43] P. Shmerkin. Projections of self-similar and related fractals: Asurvey of recent developments. Fractal Geometry and Stochastics V, (2015), vol. 70, pp. 53-74.
  * [44] P. Shmerkin and B. Solomya. Absolute continuity of self-similar measures, their projections and convolution. Trans. Amer. Math. Soc., (2016), vol. 368, pp. 5125-5151.
  * [45] B. Selmi. A note on the effect of projections on both measures and the generalization of $q$-dimension capacity. Probl. Anal. Issues Anal., (2016), vol. 5 (23), pp. 38-51.
  * [46] B. Selmi. On the effect of projections on the Billingsley dimensions. Asian-Eur. J. Math., 2020, vol. 13, pp. 2050128, 1-17.
  * [47] B. Selmi. Multifractal dimensions for projections of measures. Bol. Soc. Paran. Mat., (to appear). https://doi:10.5269/bspm.44913
  * [48] B. Selmi. _On the projections of the multifractal packing dimension for $q>1$_. Ann. Mat. Pura Appl., (to appear). https://doi:10.1007/s10231-019-00929-7
  * [49] B. Selmi and N. Yu. Svetova. On the projections of mutual $L^{q,t}$-spectrum. Probl. Anal. Issues Anal., (2017), vol. 6 (24), pp. 94 - 108.
  * [50] O. Zindulka. Packing measures and dimensions on Cartesian products. Publ. Mat., (2013), vol. 57, pp. 393-420.

Bilel SELMI

Analysis, Probability and Fractals Laboratory LR18ES17
Faculty of Sciences of Monastir
Department of Mathematics
University of Monastir
5000-Monastir
Tunisia

E-mail<EMAIL_ADDRESS>


# Sunny and Dark Outside?! Improving Answer Consistency
in VQA through Entailed Question Generation

1Arijit Ray, 1Karan Sikka, 1Ajay Divakaran, *Stefan Lee, 1Giedrius Burachas
<EMAIL_ADDRESS><EMAIL_ADDRESS>
1SRI International, *Georgia Institute of Technology

###### Abstract

While models for Visual Question Answering (VQA) have steadily improved over
the years, interacting with one quickly reveals that these models lack
consistency. For instance, if a model answers “red” to “What color is the
balloon?”, it might answer “no” if asked, “Is the balloon red?”. These
responses violate simple notions of entailment and raise questions about how
effectively VQA models ground language. In this work, we introduce a dataset,
ConVQA, and metrics that enable quantitative evaluation of consistency in VQA.
For a given observable fact in an image (e.g. the balloon’s color), we
generate a set of logically consistent question-answer (QA) pairs (e.g. Is the
balloon red?) and also collect a human-annotated set of common-sense based
consistent QA pairs (e.g. Is the balloon the same color as tomato sauce?).
Further, we propose a consistency-improving data augmentation module, a
Consistency Teacher Module (CTM). CTM automatically generates entailed (or
similar-intent) questions for a source QA pair and fine-tunes the VQA model if
the VQA’s answer to the entailed question is consistent with the source QA
pair. We demonstrate that our CTM-based training improves the consistency of
VQA models on the ConVQA datasets and is a strong baseline for further
research.

##  1 Introduction

> “A skeptic, I would ask for consistency first of all.”
>
> Sylvia Plath Plath (2007)

Visual Question Answering (VQA) Antol et al. (2015) involves answering natural
language questions about images. Despite the recent progress on VQA, we
observe that existing methods are prone to making blatant mistakes while
answering questions regarding the same visual fact but from slightly different
perspectives (Figure 1). This reveals a critical limitation of the state-of-
the-art models in maintaining consistency.

Figure 1: Current VQA models often fail at consistently answering semantically
rephrased questions. To address this limitation, we construct a consistent VQA
(ConVQA) dataset with diverse QA pairs that query the same visual fact. We
also propose a Consistency Teacher Module (CTM) that improves VQA consistency
by rewarding consistent behavior.

In particular, we motivate our definition of consistency based on classical
deductive logic Tarski and Tarski (1994) that defines a consistent theory as
one that does not entail a contradiction. Correspondingly, we define
consistency, in the context of VQA, as being able to answer questions posed
from different semantic perspectives about a certain fact without any
contradiction. In addition, consistent Question-Answer (QA) pairs can be
derived based on simple notions of logic or by commonsense reasoning. For
instance, say an image contains a “large building”. Logic-based QA pairs can
be “is the building small? no” and “what size is the building? large”. On the
other hand, if an image contains “vegetarian pizza”, commonsense-based QA
pairs can be “is it a vegetarian pizza? yes” and “is there pepperoni on the
pizza? no”, which requires commonsense knowledge that “pepperoni” is not
vegetarian.

While attempts have been made to construct logic-based consistent VQA datasets
Hudson and Manning (2019), they still fall short on commonsense-based
consistency. To this end, our ConVQA Dataset consists of two subsets: 1) a
challenging human-annotated set comprised of commonsense-based consistent QA’s
(shown in Figure 3), and 2) an automatically generated logic-based consistent
QA dataset (shown in Figure 2).

To improve the consistency of VQA models, we propose a Consistency Teacher
Module (CTM), which consists of a Question Generator that synthesizes entailed
(or similar-intent) questions given a seed QA pair and a Consistency Checker
that examines whether answers to those similar-intent questions are
consistent. For training a consistent VQA model, our CTM acts as a
consistency-based data augmentation scheme that trains a VQA model with
consistent answers to entailed questions. We demonstrate that our approach
improves the performance of a baseline VQA model on our ConVQA testing sets in
terms of both accuracy and consistency. Our datasets and models will be
available at https://bit.ly/32exlM7.

Figure 2: Qualitative examples from our ConVQA dataset derived based on logic.
Figure 3: Qualitative examples from our human-annotated commonsense-based
ConVQA Dataset. Figure 4: Block diagram of the proposed CTM including a
Question Generator that synthesizes questions with similar intent and a
Consistency Checker that classifies QA pairs as consistent, unrelated, or
contradictory. CTM finetunes VQA models via reinforcement learning with the
answer consistency as the reward to encourage VQA models to answer rephrased
questions more consistently. The examples shown are from a real run.

##  2 Related Work

Checking for consistency can be considered as an interrogative Turing Test
Radziwill and Benton (2017) for linguistic robustness Stede (1992), Li et al.
(2017). Works such as Xu et al. (2018) explore the robustness of VQA with
respect to image variations, whereas works such as Ray et al. (2016) and
Mahendru et al. (2017) focus on the understanding of the premise of a question
instead of relying on dataset biases Agrawal et al. (2017) Goyal et al. (2017)
or linguistic biases Ramakrishnan et al. (2018).

Recently, the research community has shown great interest in evaluating VQA
for consistency and plausibility. GQA Hudson and Manning (2019) is established
as a scene-graph based QA dataset. Their questions, similar to Johnson et al.
(2017), require multiple hops of reasoning, and are not validated or annotated
by humans. Our ConVQA differs from GQA in the following two aspects. First, we
provide a human-validated test set of the automatically generated logic-based
consistent QA’s for a more accurate performance evaluation. Second, we collect
human-annotated QA pairs based on common-sense in addition to the logic-based
QA’s. The most relevant work to ours is Shah et al. (2019). However, they
focus strictly on question paraphrases that maintains the same answers as the
source question. We, however, focus on generating questions which can have
different answers, but are about the same visual fact, which greatly increases
the diversity of the resulting QA pairs. To the best of our knowledge, the
proposed ConVQA dataset is the first consistent QA dataset that contains
human-annotated consistent QA’s based on common-sense.

Other works have also looked into question generation Zhang et al. (2016),
Mostafazadeh et al. (2016) for training better VQA models. In Misra et al.
(2017), QA pairs are obtained from an oracle in a simulated environment. In
contrast, our CTM-based training operates on real images and uses a learned
consistency measure to train the VQA module with consistent QA’s.

##  3 ConVQA Datasets

The consistent QA pairs in our ConVQA are generated automatically based on
simple notions of logical consistency or are human-annotated using commonsense
reasoning.

Logic-based Consistent QA. (L-ConVQA) Consider the Visual Genome Krishna et
al. (2017) scene graph in Figure 2 consisting of objects, attributes, and
their relationships. We consider each triplet to encode a single ‘visual
fact’, for instance, that the sofa is white. We employ slot-filler NLP
techniques to generate a set of QA pairs for each triplet (object-relation-
subject) in the scene graph. Currently, we focus on attribute (e.g., color,
size), existential (e.g., is there) and relational (e.g., sofa on floor)
consistency. We leverage Wordnet Miller (1995) and a manually generated list
of antonyms (e.g., white vs. black) and hypernyms (e.g., white $\rightarrow$
color) to generate these QA pairs. For example, for the attribute “white” of
an object “cup”, we generate QA pairs such as “is the cup white? yes”, “is the
cup black? no” and “what color is cup? white”. We also filter objects and
relationships by frequency and saliency (e.g., based on bounding boxes) to
avoid non-salient and infrequent objects or noisy relationships. We have a
total of 880,141 QA pairs in 255,910 sets on 70,292 images. We split the data
into a training set with 47,999 images, a validation set with 9,993 images,
and a test set with 12,300 images. Notably, we create a smaller clean test set
(12,325 QA pairs on 725 images) using Amazon Mechanical Turk (AMT) where three
independent workers were asked to remove incorrect or unnatural QA’s.

Commonsense-based Consistent QA. While the logic-based consistent QA set
provides a first step into large-scale examination of VQA consistency, the
generated questions require limited reasoning and commonsense and are,
therefore, frequently simpler than human-annotated ones. Hence, we collect
more challenging QA pairs based on commonsense (CS-ConVQA) by asking AMT
workers to write intelligent rephrases of QA pairs sampled from the VQA2.0
Goyal et al. (2017) validation dataset. AMT workers were instructed to avoid
simple word paraphrases and instead to write rephrases that require
commonsense reasoning in order to answer the question consistently. We collect
approximately 3.5 consistent QA pairs per image for 6439 images. After
filtering images that overlap with the training set of the L-ConVQA subset, we
split this data into a training set (1568 images), a validation set (450
images), and a test set (1590 images).

##  4 Approach

To improve VQA consistency, we propose training a VQA model using a
Consistency Teacher Module (CTM) that generates entailed questions and
performs a consistency-based data augmentation. More specifically, CTM
consists of two trainable components – an entailed question generator and a
consistency checker.

Entailed Question Generator. For a given a source question-answer pair, we
define entailed questions as those for which the answer should be obvious
given the source QA pair. For example, given the source QA pair “Who is on
court? Man”, an appropriate entailed question might be “Is the tennis court
empty?”. We train a question generation model that given representations of
the image and the source QA pair, generates a new question. Specifically, our
question generator concatenates the deep features of an image (extracted using
a ResNet152 He et al. (2016) network) and a QA pair (extracted using a 1-layer
LSTM Hochreiter and Schmidhuber (1997)) to represent a visual fact. These
features are fed into another LSTM model to generate a similar-intent
question. We train this module on the automatically generated Logical L-ConVQA
train set. We also include some closely related (according to averaged
Word2Vec Mikolov et al. (2013) distance) Visual Genome Krishna et al. (2017)
QA pairs in the training of the question generator to add some diversity to
the generated questions.

Consistency Checker. Once the VQA model produces an answer for the generated
entailed question, it may or may not be consistent with the source question.
To evaluate this and provide feedback to the VQA model, we train a consistency
checker that processes the image and both the source and entailed QA pairs.
Similar to the question generator architecture, the consistency checker takes
in deep features of the image and two QA pairs and classifies the QA pairs as
consistent, inconsistent or unrelated. This model is trained using the
automatically generated L-ConVQA train set alone. Inconsistent examples in the
L-ConVQA set are made using simple techniques such as flipping yes/no answers
and replacing entities in the scene graph triplets.

Consistency Teacher Module (CTM). Putting these two components together, we
can train a VQA model based on its consistency on generated entailed
questions. Figure 4 shows our pipeline. During training, for each source VQA
QA pair (“Who is on court? Man”), we generate an entailed question (“Is tennis
court empty?”) and produce the VQA model’s answer (“No”). We then run the
consistency checker to determine if the generated answer is consistent with
the source QA (in this case, “Who is on court? Man”). If the answer is
consistent (and VQA confidence $>0.7$), we treat it as the ground truth for
the entailed question and update the VQA model as if this example were part of
the original dataset. Likewise if it is deemed inconsistent, or if the
question is deemed unrelated, it is unclear what the correct answer should be,
so we do not update the model.

##  5 Experiments

Figure 5: Qualitative examples of entailed question generation and consistency
checking for training the VQA. Table 1: Performance comparison of baseline VQA
trained on VQA2.0, baseline VQA finetuned on ConVQA, and VQA trained using our
CTM. L-ConVQA is the human-cleaned Logical Consistent QA dataset, CS-ConVQA is
the human annotated Common-sense Consistency Dataset and VG is Visual Genome.
CTM-based training produces the best results in terms of overall accuracy and
consistency. DATA denotes the data used to fine-tune VQA or seed the CTM
question generator.

|  | DATA | L-ConVQA |  | CS-ConVQA
---|---|---|---|---|---
|  |  | Perf Con | Avg Con | Top1 |  | Perf Con | Avg Con | Top1 | Yes/No | Num
a) | VQA | VQA2.0 | 36.25 | 71.36 | 70.34 |  | 26.13 | 59.61 | 60.03 | 65.49 | 31.39
b) | FineTune | CS-ConVQA | 34.54 | 70.39 | 69.48 |  | 26.39 | 59.65 | 60.07 | 65.80 | 35.92
c) | FineTune | L/CS-ConVQA | 54.68 | 83.42 | 83.16 |  | 24.70 | 59.30 | 59.60 | 65.14 | 33.33
d) | +CTM | L/CS-ConVQA | 54.6 | 83.23 | 82.79 |  | 25.94 | 60.39 | 60.78 | 66.63 | 36.89
e) | FineTune | L/CS-ConVQA,VG | 36.40 | 71.60 | 70.94 |  | 25.22 | 59.19 | 59.56 | 65.30 | 31.39
f) | +CTMvg | L/CS-ConVQA,VG | 51.41 | 81.66 | 81.37 |  | 27.49 | 59.75 | 60.15 | 66.41 | 34.95

Figure 6: Qualitative examples of our improved VQA consistency by CTM training
compared to baseline bottom-up VQA. Green is correct, Red is wrong. GT is
Ground Truth.

To evaluate our approach, we apply the Consistency Teacher Module (CTM) module
to a state-of-the-art VQA model trained on VQAv2 and evaluate performance on
the ConVQA datasets. We describe training procedures, metrics, and baselines
in this section.

Consistency Teacher Module Training. We train the components of CTM – Entailed
Question Generation and Consistency Checker – using only the synthetic
L-ConVQA train set (referred as the standard CTM) or a mix of Visual Genome
and L-ConVQA train (referred as CTMvg) and keep them frozen when fine-tuning
the VQA model. When we train the VQA, the sets used to finetune the VQA or
seed the CTM come from splits not seen during training of the CTM - val split
of L-ConVQA and train split of CS-ConVQA.

Qualitative examples of our Entailed Question Generator trained only on
L-ConVQA are shown in Figure 5. Despite only being trained on the
automatically-generated L-ConVQA data, it generates reasonably well-entailed
questions on human-annotated questions.

Our Consistency Checker has a high accuracy of classification on the L-ConVQA
test set ($90\%$). However, when tested with a mix of commonsense-based CS-
ConVQA, the accuracy drops to $64\%$ (chance is $33\%$ for $3$ classes). We
find that precision is important when training the VQA using the pre-trained
Consistency Checker. Hence, we use the classifier at above $90\%$ confidence
threshold, where the precision is $70.38\%$.

Evaluation Metrics for ConVQA. We report three metrics for ConVQA – capturing
notions of consistency and performance.

  1. –

Perfect-Consistency (Perf-Con). A model is perfectly consistent for a question
set if it answers all questions in the set correctly. We report the percentage
of such sets as Perf-Con.

  2. –

Average Consistency (Avg-Con). We also report the average accuracy within a
consistent question set over the entire dataset as Avg-Con.

  3. –

Accuracy (top-1). Finally, we report the top-1 accuracy over all questions in
the dataset.

Baselines. We compare to a number of baseline models to put our CTM results
into context:

  1. –

VQA. We take the bottom-up top-down VQA model Anderson et al. (2018) as our
base model for these experiments. To evaluate consistency in existing models,
we present results on ConVQA of this model pretrained on VQA2.0.

  2. –

Finetuned models: We present results for models finetuned on ConVQA and Visual
Genome – Finetune CS-ConVQA finetuned on the commonsense ConVQA dataset,
Finetune L/CS-ConVQA on both logical and commonsense ConVQA, and Finetune
L/CS-ConVQA,VG extending to Visual Genome questions.

When we apply our CTM model to the finetuned baselines above, we seed the
question generator with the associated dataset.

##  6 Results and Analysis

Table 1 shows quantitative results on our L-ConVQA and CS-ConVQA datasets. We
make a number of observations below.

The state-of-the-art VQA has low consistency. The baseline VQA system (row a)
retains similarly high top-1 accuracy on the ConVQA splits (63.58% on VQAv2 vs
70.34% / 60.03% on L-ConVQA / CS-ConVQA); however, it achieves only 26.13%
perfect consistency on the human generated CS-ConVQA questions.

Finetuning is an effective strategy for the synthetic L-ConVQA split.
Finetuning on L-ConVQA train results in 18.43% gains in perfect consistency on
L-ConVQA test (row c vs a). This is unsurprising given the templated questions
and simple concepts in L-ConVQA; however, perfect consistency is low in
absolute terms at 54.68%.

Finetuning does not lead to significant gains in consistency for human-
generated questions. Finetuning the VQA model on CS-ConVQA (row b) leads to an
improvement in consistency of only 0.26%. Likewise, adding L-ConVQA (row c)
and extra Visual Genome questions (row e) actually reduces consistency.

CTM-based training preserves or improves consistency when leveraging
additional data. When we apply CTM to the Finetuned L/CS-ConVQA model, we
improve CS-ConVQA perfect consistency by 1.24% (row d vs c) while modestly
improving other metrics. Extending to Visual Genome questions, the CTM
augmented model improves perfect consistency in CS-ConVQA by 2.27% over the
finetuned model (row f vs e). Interestingly, the CTM modules were never
trained with the human-annotated CS-ConVQA questions and yet lead to this
improvement on CS-ConVQA by acting as an intelligent data
augmenter/regularizer.

##  7 Conclusion and Discussion

In this paper, we introduced a ConVQA dataset consisting of logic-based and
commonsense-based consistent QA pairs about visual facts in an image. We also
proposed a Consistency Teacher Module that acts as a consistency-based data
augmenter to teach VQA models to answer consistently. As future work, we plan
to look into improving our automatically generated consistent QA pairs using
external knowledge-bases.

## Acknowledgements

We thank Yi Yao, Xiao Lin, Anirban Roy, Avi Ziskind and the anonymous
reviewers for their helpful comments. This research was developed under DARPA
XAI. The views, opinions and/or findings expressed are those of the authors’
and should not be interpreted as representing the official views/policies of
the DoD / U.S. Govt.

## References

  * Agrawal et al. (2017) Aishwarya Agrawal, Aniruddha Kembhavi, Dhruv Batra, and Devi Parikh. 2017.  C-vqa: A compositional split of the visual question answering (vqa) v1. 0 dataset.  _arXiv preprint arXiv:1704.08243_.
  * Anderson et al. (2018) Peter Anderson, Xiaodong He, Chris Buehler, Damien Teney, Mark Johnson, Stephen Gould, and Lei Zhang. 2018.  Bottom-up and top-down attention for image captioning and visual question answering.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 6077–6086.
  * Antol et al. (2015) Stanislaw Antol, Aishwarya Agrawal, Jiasen Lu, Margaret Mitchell, Dhruv Batra, C Lawrence Zitnick, and Devi Parikh. 2015.  Vqa: Visual question answering.  In _Proceedings of the IEEE International Conference on Computer Vision_ , pages 2425–2433.
  * Goyal et al. (2017) Yash Goyal, Tejas Khot, Douglas Summers-Stay, Dhruv Batra, and Devi Parikh. 2017\.  Making the v in vqa matter: Elevating the role of image understanding in visual question answering.  In _CVPR_ , volume 1, page 3.
  * He et al. (2016) Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016.  Deep residual learning for image recognition.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 770–778.
  * Hochreiter and Schmidhuber (1997) Sepp Hochreiter and Jürgen Schmidhuber. 1997.  Long short-term memory.  _Neural computation_ , 9(8):1735–1780.
  * Hudson and Manning (2019) Drew A Hudson and Christopher D Manning. 2019.  Gqa: a new dataset for compositional question answering over real-world images.  _arXiv preprint arXiv:1902.09506_.
  * Johnson et al. (2017) Justin Johnson, Bharath Hariharan, Laurens van der Maaten, Li Fei-Fei, C Lawrence Zitnick, and Ross Girshick. 2017.  Clevr: A diagnostic dataset for compositional language and elementary visual reasoning.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 2901–2910.
  * Krishna et al. (2017) Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalantidis, Li-Jia Li, David A Shamma, et al. 2017\.  Visual genome: Connecting language and vision using crowdsourced dense image annotations.  _International Journal of Computer Vision_ , 123(1):32–73.
  * Li et al. (2017) Yitong Li, Trevor Cohn, and Timothy Baldwin. 2017.  Robust training under linguistic adversity.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers_ , pages 21–27.
  * Mahendru et al. (2017) Aroma Mahendru, Viraj Prabhu, Akrit Mohapatra, Dhruv Batra, and Stefan Lee. 2017\.  The promise of premise: Harnessing question premises in visual question answering.  _arXiv preprint arXiv:1705.00601_.
  * Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. 2013.  Distributed representations of words and phrases and their compositionality.  In _Advances in neural information processing systems_ , pages 3111–3119.
  * Miller (1995) George A Miller. 1995.  Wordnet: a lexical database for english.  _Communications of the ACM_ , 38(11):39–41.
  * Misra et al. (2017) Ishan Misra, Ross Girshick, Rob Fergus, Martial Hebert, Abhinav Gupta, and Laurens van der Maaten. 2017.  Learning by asking questions.  _arXiv preprint arXiv:1712.01238_.
  * Mostafazadeh et al. (2016) Nasrin Mostafazadeh, Ishan Misra, Jacob Devlin, Margaret Mitchell, Xiaodong He, and Lucy Vanderwende. 2016.  Generating natural questions about an image.  _arXiv preprint arXiv:1603.06059_.
  * Plath (2007) Sylvia Plath. 2007.  _The unabridged journals of Sylvia Plath_.  Anchor.
  * Radziwill and Benton (2017) Nicole M Radziwill and Morgan C Benton. 2017.  Evaluating quality of chatbots and intelligent conversational agents.  _arXiv preprint arXiv:1704.04579_.
  * Ramakrishnan et al. (2018) Sainandan Ramakrishnan, Aishwarya Agrawal, and Stefan Lee. 2018.  Overcoming language priors in visual question answering with adversarial regularization.  In _Advances in Neural Information Processing Systems_ , pages 1541–1551.
  * Ray et al. (2016) Arijit Ray, Gordon Christie, Mohit Bansal, Dhruv Batra, and Devi Parikh. 2016.  Question relevance in vqa: identifying non-visual and false-premise questions.  _arXiv preprint arXiv:1606.06622_.
  * Shah et al. (2019) Meet Shah, Xinlei Chen, Marcus Rohrbach, and Devi Parikh. 2019.  Cycle-consistency for robust visual question answering.  _arXiv preprint arXiv:1902.05660_.
  * Stede (1992) Manfred Stede. 1992.  The search for robustness in natural language understanding.  _Artificial Intelligence Review_ , 6(4):383–414.
  * Tarski and Tarski (1994) Alfred Tarski and Jan Tarski. 1994.  _Introduction to Logic and to the Methodology of the Deductive Sciences_.  24\. Oxford University Press on Demand.
  * Xu et al. (2018) Xiaojun Xu, Xinyun Chen, Chang Liu, Anna Rohrbach, Trevor Darrell, and Dawn Song. 2018.  Fooling vision and language models despite localization and attention mechanism.  In _Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition_ , pages 4951–4961.
  * Zhang et al. (2016) Shijie Zhang, Lizhen Qu, Shaodi You, Zhenglu Yang, and Jiawan Zhang. 2016.  Automatic generation of grounded visual questions.  _arXiv preprint arXiv:1612.06530_.