# Large sieve inequalities with power moduli and Waring’s problem

Stephan Baier and Sean B. Lynch

##  1 Introduction

Let $M$ be an integer, let $N$ be a positive integer, and let $\\{z_{n}\\}$ be
a sequence of complex numbers. Let $\mathcal{S}$ be a finite set of real
numbers, and let $\delta\leq 1$ be a positive real number. We write
$\lVert\alpha\rVert$ for the distance from a real number $\alpha$ to its
nearest integer, and we also write $e(\alpha)=e^{2\pi i\alpha}$. Moreover, we
say that the set $\mathcal{S}$ is $\delta$-spaced modulo 1 if
$\lVert\alpha-\beta\rVert\geq\delta$ whenever $\alpha$ and $\beta$ are
distinct members of $\mathcal{S}$. The general large sieve inequality says
that if $\mathcal{S}$ is $\delta$-spaced modulo 1, then

$\displaystyle\sum_{\alpha\in\mathcal{S}}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq(\delta^{-1}+N-1)\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$
(1)

Montgomery wrote a lovely expository article on this result [Mon78].

Let $Q$ and $k$ be positive integers. Then we take the set

$\displaystyle\mathcal{S}_{k}(Q)=\left\\{\text{$a/q^{k}:a$ and $q$ are coprime
positive integers with $a\leq q^{k}\leq Q^{k}$}\right\\}$ (2)

so that

$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}=\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}.$

This paper is about large sieve inequalities with
$k\textsuperscript{th}$-power moduli. These are inequalities of the form

$\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}\leq\Delta_{k}(Q,N)\sum_{n=M+1}^{M+N}|z_{n}|^{2},$

where $\Delta_{k}(Q,N)$ is independent of $M$ and $\\{z_{n}\\}$. We now let
$\Delta_{k}(Q,N)$ be the infimum of such constants so that our large sieve
inequalities are just upper bounds on $\Delta_{k}(Q,N)$.

In the first paper on these bounds, Zhao observed two bounds that trivially
follow from the general large sieve inequality [Zha04]. Firstly, since the set
$\mathcal{S}_{k}(Q)$ is $Q^{-2k}$-spaced modulo 1, we have

$\displaystyle\Delta_{k}(Q,N)\leq Q^{2k}+N-1.$ (3)

Secondly, we can apply the general large sieve inequality to the sum over $a$,
and then sum over $q$. This gives

$\displaystyle\Delta_{k}(Q,N)\leq\sum_{q=1}^{Q}(q^{k}+N-1)\leq Q(Q^{k}+N-1).$
(4)

If $N\gg Q^{2k}$, then bound (3) gives $\Delta_{k}(Q,N)\ll N$. If $N\ll
Q^{k}$, then bound (4) gives $\Delta_{k}(Q,N)\ll Q^{k+1}$. Combining these, we
have

$\displaystyle N\gg Q^{2k}\mbox{ or }N\ll Q^{k}\implies\Delta_{k}(Q,N)\ll
Q^{k+1}+N.$ (5)

It’s not hard to see that

$\displaystyle\Delta_{k}(Q,N)\gg Q^{k+1}+N.$ (6)

So, in particular, (5) is sharp. If $z_{M+1}=1$ and every other $z_{n}=0$,
then we compute that

$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}=|\mathcal{S}_{k}(Q)|\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$

Therefore

$\displaystyle\Delta_{k}(Q,N)\geq|\mathcal{S}_{k}(Q)|=\sum_{q=1}^{Q}\varphi(q^{k})=\sum_{q=1}^{Q}q^{k-1}\varphi(q)\gg
Q^{k+1}.$ (7)

If each $z_{n}=e(-n\alpha_{0})$, for a fixed
$\alpha_{0}\in\mathcal{S}_{k}(Q)$, then we have

$\sum_{\alpha\in\mathcal{S}_{k}(Q)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\geq\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha_{0})\right|^{2}=N\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$

Therefore

$\displaystyle\Delta_{k}(Q,N)\geq N.$ (8)

Combining bounds (7) and (8), we get bound (6).

In the case $k=1$, bounds (3) and (6) collectively show that
$\Delta_{1}(Q,N)\asymp Q^{2}+N$. Otherwise, we call

$\displaystyle Q^{k}<N<Q^{2k}$ (9)

the nontrivial range. This is where many authors [Zha04, BZ05, Bai06, BZ08,
Hal12, Hal15, Hal18, Hal20, Mun21, BMS22, McG22] beat the trivial bounds and
worked toward Zhao’s conjecture that

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}(Q^{k+1}+N).$

Baier, Lynch and Zhao showed that the factor of $Q^{\varepsilon}$ cannot be
removed when $k=2$ [BLZ19]. Their method extends to all $k\geq 2$.

Our method of attacking the nontrivial range is simple, effective and can
easily be applied to more general large sieve inequalities with polynomial
moduli. We choose to illustrate our method in this restricted setting of
$k\textsuperscript{th}$-power moduli for three reasons. Firstly, these large
sieve inequalities have received a lot of attention, and we achieve savings in
substantial ranges for all $k\geq 4$. Secondly, they have many (actualised and
potential) applications, as listed in the introduction to [BMS22]. Lastly, our
method relates these large sieve inequalities to Waring’s problem with
$k\textsuperscript{th}$-powers.

For an introduction to Vinogradov’s mean value theorem and its application to
Waring’s problem, look to Vaughan’s book [Vau97, Ch. 5]. In the case $k=3$,
Wooley proved a long standing conjectured bound on Vinogradov mean values
[Woo16, Thm. 1.1]. Bourgain, Demeter and Guth were the first to prove the
conjectured bound for all $k\geq 3$ [BDG16, Thm. 1.1]. We make use of Wooley’s
generalised version [Woo18, Thm. 1.1] of Bourgain, Demeter and Guth’s bound.
See [Coo+22] for an interesting comparison between the method of Wooley and
that of Bourgain, Demeter and Guth.

##  2 Results

Let $Q,N,n$ and $k,s,t$ be positive integers. In light of the introduction, we
restrict to the nontrivial range

$\displaystyle Q^{k}<N<Q^{2k}.$

Let $R_{k,s}(n)$ be the number of $s$-tuples of positive integers
$(n_{1},\ldots,n_{s})$ satisfying $n=n_{1}^{k}+\ldots+n_{s}^{k}$. The
forthcoming implied constants are independent of $Q,N,n$. They may depend on
$k,s,t$ or on an arbitrarily small positive real number $\varepsilon$.

We pass to counting Farey fractions in short intervals in Section 4. This is
standard practice. We then lay the foundations of our new method in Section 5,
where we also prove the following two lemmata.

######  Lemma 2.1.

Suppose that, uniformly in positive integers $n\leq sQ^{k}$, we have the bound

$\displaystyle R_{k,s}(n)\ll C_{\varepsilon,k,s}(Q).$

Then we have

$\displaystyle\Delta_{k}(Q,N)\ll
NQ^{\varepsilon}\left(C_{\varepsilon,k,s}(Q)\cdot N^{-1}Q^{2k}\right)^{1/s}.$

######  Lemma 2.2.

Suppose that, uniformly in subsets $\mathcal{A}$ of positive integers $\leq
Q$, we have the bound

$\displaystyle\int_{0}^{1}\left|\sum_{n\in\mathcal{A}}e(\alpha
n^{k})\right|^{2s}d\alpha\ll D_{\varepsilon,k,s}(Q)\cdot|\mathcal{A}|^{s}.$
(10)

Then we have

$\displaystyle\Delta_{k}(Q,N)\ll
NQ^{\varepsilon}\left(D_{\varepsilon,k,s}(Q)\cdot N^{-1}Q^{2k}\right)^{1/s}.$

In Section 6, we recall an elementary divisor bound for $R_{k,2}(n)$. Running
this through Lemma 2.1 with $s=2$ recovers Baier and Zhao’s large sieve
inequality [BZ05, Thm. 1.1]. We state this large sieve inequality in part (i)
of our first theorem.

Also in Section 6, we recall Marmon’s non-elementary bound on $R_{k,4}(n)$
[Mar10, Thm. 1.3]. Marc Munsch suggested that we insert this into Lemma 2.1
with $s=4$ so as to recover Baker, Munsch and Shparlinski’s large sieve
inequality [BMS22, Thm. 1.4]. We state this large sieve inequality in part
(ii) of our first theorem. In fact, we state the inequality for all $k\geq 3$,
while Baker, Munsch and Shparlinski left a void when $k=4$. McGrath [McG22,
Cor. 1.4] filled this void, albeit with a weaker bound than the general
formula in $k$ would suggest. Our result corrects this, filling the void with
the appropriate bound.

######  Theorem 2.3.

  1. (i)

For all $k\geq 2$, we have the large sieve inequality

$\displaystyle\Delta_{k}(Q,N)\ll N^{\frac{1}{2}}Q^{k+\varepsilon}.$

  2. (ii)

For all $k\geq 3$, we have the large sieve inequality

$\displaystyle\Delta_{k}(Q,N)\ll
N^{\frac{3}{4}}Q^{\frac{k}{2}+\frac{1}{4}+\frac{1}{2\sqrt{k}}+\varepsilon}.$

Finally, in Section 7, we invoke a recent and high-powered theorem of Wooley
[Woo18, Thm. 1.1]. We end up with a family, indexed by positive integers
$t\leq k$, of mean value estimates of the form (10). Running this through
Lemma 2.2 with $s=\frac{1}{2}t(t+1)$ results in our second theorem.

######  Theorem 2.4.

For any positive integer $t\leq k$, we have the large sieve inequality

$\displaystyle\Delta_{k}(Q,N)\ll
N^{1-\frac{2}{t(t+1)}}Q^{1+\frac{4k-2t}{t(t+1)}+\varepsilon}.$ (11)

######  Remark 2.5.

Write $N=Q^{\lambda}$ for a real $\lambda$ with $k<\lambda<2k$. Then the
right-hand side of (11) becomes

$\displaystyle Q^{f_{k,t}(\lambda)+\varepsilon},$

where

$\displaystyle
f_{k,t}(\lambda)=\lambda\left(1-\frac{2}{t(t+1)}\right)+1+\frac{4k-2t}{t(t+1)}=\lambda+1+\frac{2(2k-\lambda)}{t(t+1)}-\frac{2}{t+1}.$

We minimise $f_{k,t}(\lambda)$ by taking

$t=\min\\{\lceil 2(2k-\lambda)\rceil,k\\}.$

Incidentally, this choice of $t$ makes $f_{k,t}(\lambda)$ continuous in
$\lambda$.

As we’ll see in the next section, Theorem 2.4 with $t=k$ is particularly
powerful.

######  Corollary 2.6.

We have the large sieve inequality

$\displaystyle\Delta_{k}(Q,N)\ll
N^{1-\frac{2}{k(k+1)}}Q^{1+\frac{2}{k+1}+\varepsilon}.$

To discuss a conditional result, we assume that $k\geq 3$ and $s\geq k+1$
throughout the rest of this section. Recall the Hardy-Littlewood singular
series

$\displaystyle\mathfrak{S}_{k,s}(n)=\sum_{q=1}^{\infty}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q}\left(\frac{1}{q}\sum_{r=1}^{q}e(ar^{k}/q)\right)^{s}e(-na/q).$

Then Hardy and Littlewood proved that whenever $s$ is sufficiently large
relative to $k$, we have the formula

$\displaystyle
R_{k,s}(n)=\frac{\Gamma(1+\frac{1}{k})^{s}}{\Gamma(\frac{s}{k})}\mathfrak{S}_{k,s}(n)n^{\frac{s}{k}-1}+o(n^{\frac{s}{k}-1}).$
(12)

See [Vau97, Ch. 2] for a modern proof of this formula, with $s>2^{k}$ and a
power saving in the error term. We are often interested in obtaining
$\mathfrak{S}_{k,s}(n)\gg 1$ so that the first term on the right-hand side is
not swallowed by the error, thereby making (12) a fully-fledged asymptotic
formula. This lower-bound on the singular series can be subtle, as covered in
[Vau97, §. 4.5]. Fortunately, for our application, the only bound we need on
the singular series is a modest upper-bound. Indeed, we have [Vau97, Thm. 4.3,
p. 49]

$\mathfrak{S}_{k,s}(n)\ll n^{\varepsilon}.$

Therefore, if formula (12) holds, then we have

$\displaystyle R_{k,s}(n)\ll n^{\frac{s}{k}-1+\varepsilon}.$ (13)

If bound (13) holds, then it follows from Lemma 2.1 that

$\displaystyle\Delta_{k}(Q,N)\ll
N^{1-\frac{1}{s}}Q^{1+\frac{k}{s}+\varepsilon}.$ (14)

Rewriting this large sieve bound as $NQ^{1+\varepsilon}(N^{-1}Q^{k})^{1/s}$
and recalling that $N>Q^{k}$, it’s optimal to choose $s$ as small as possible.
As luck would have it, this runs with – not against – the difficulty in
Waring’s problem! The Hardy-Littlewood asymptotic formula (12) is conjectured
to hold with $s=k+1$ whenever $k\geq 3$. If this famous conjecture holds, then
large sieve bound (14) holds with $s=k+1$ whenever $k\geq 3$.

##  3 Comparison with previous results

Let $Q,N,k$ be positive integers. The forthcoming implied constants are
independent of $Q$ and $N$, but they may depend on $k$ or on an arbitrarily
small positive real number $\varepsilon$.

The standard way of comparing bounds is to write $N=Q^{\lambda}$ for a real
parameter $\lambda$. Restricting to the nontrivial range (9), we assume that
$k<\lambda<2k$. It is also customary to introduce clean notation for some
functions of $k$ that appear in exponents. Put

$\displaystyle\kappa=\frac{1}{2^{k-1}}\qquad\mbox{and}\qquad\omega=\frac{1}{(k-1)(k-2)+2}.$

###  3.1 $k\geq 5$

In this subsection, we assume that $k\geq 5$. Here is the list of bounds with
which we have to contend:
First trivial (3)

$\displaystyle\Delta_{k}(Q,N)\ll Q^{2k}+N,$ (15)

Second trivial (4)

$\displaystyle\Delta_{k}(Q,N)\ll Q(Q^{k}+N),$ (16)

Zhao [Zha04]

$\displaystyle\Delta_{k}(Q,N)\ll(Q^{k+1}+NQ^{1-\kappa}+N^{1-\kappa}Q^{1+k\kappa})Q^{\varepsilon},$
(17)

Baier and Zhao [BZ05]

$\displaystyle\Delta_{k}(Q,N)\ll(Q^{k+1}+N+N^{\frac{1}{2}}Q^{k})Q^{\varepsilon},$
(18)

Halupczok [Hal12, Hal15, Hal18, Hal20]

$\displaystyle\Delta_{k}(Q,N)\ll\left(Q^{k+1}+\min\left\\{N^{1-\omega}Q^{1+(2k-1)\omega},NQ^{1-\frac{1}{k(k-1)}}+N^{1-\frac{1}{k(k-1)}}Q^{\frac{k}{k-1}}\right\\}\right)Q^{\varepsilon},$
(19)

Munsch [Mun21]

$\displaystyle\Delta_{k}(Q,N)\ll
N^{1-\frac{1}{k(k+1)}}Q^{1+\frac{1}{k+1}+\varepsilon},$ (20)

Baker, Munsch and Shparlinski [BMS22]

$\displaystyle\Delta_{k}(Q,N)\ll
NQ^{\frac{1}{2}}+N^{\frac{3}{4}}Q^{\frac{k}{2}+\frac{1}{4}+\frac{1}{2\sqrt{k}}+\varepsilon}.$
(21)

Corollary 2.6 is our main bound. It is sharper than all of the previous bounds
for $\lambda<\lambda_{0}$, where $\lambda_{0}=2k-3+O(k^{-\frac{1}{2}})$ and
$2k-3<\lambda_{0}<2k-1$. As $k$ tends to infinity, our main bound tends to
improve $100\%$ of the nontrivial range. In particular, when $k=5$, it already
improves more than $69\%$ of the nontrivial range. This percentage is more
than $86\%$ when $k=16$, and it is more than $97\%$ when $k=100$.

For the remaining range $\lambda\geq\lambda_{0}$, (18) and (21) are the
strongest of the previous bounds. Theorem 2.3 recovers both of these, and the
proofs we give are simpler than the previous proofs. Most notably, Marc Munsch
observed that we may completely bypass Baker, Munsch and Shparlinski’s route
through asymmetric additive energy; they also start with Marmon’s bound on
$R_{k,4}(n)$ to prove (21).

By applying Theorem 2.4 with smaller values of $t$, we increase the range in
which we improve on these two previous bounds. Indeed, Theorem 2.4 with $t=3$
(or even $t=4$) beats (18) for $\lambda<2k-\frac{3}{2}$. The choice of $t$
that maximises our range of improvement on (21) changes with $k$. When $5\leq
k<16$, we take $t=3$ to improve on (21) in the range
$\lambda<2k-3+\frac{6}{\sqrt{k}}$. When $16\leq k<100$, we take $t=4$ to
improve on (21) in the range $\lambda<2k-\frac{7}{3}+\frac{10}{3\sqrt{k}}$.
Finally, when $k\geq 100$, we take $t=5$ to improve on (21) in the range
$\lambda<2k-\frac{25}{11}+\frac{30}{11\sqrt{k}}$.

Now let

$\lambda_{1}=\begin{cases}2k-\frac{3}{2}&\mbox{ if }5\leq k<16\\\
2k-\frac{7}{3}+\frac{10}{3\sqrt{k}}&\mbox{ if }16\leq k<100\\\
2k-\frac{25}{11}+\frac{30}{11\sqrt{k}}&\mbox{ if }k\geq 100.\end{cases}$

Then Theorem 2.4, taken as one whole, beats all of the previous bounds for
$\lambda<\lambda_{1}$. So Theorem 2.4 improves $70\%$ of the nontrivial range
when $k=5$. This percentage is more than $90\%$ when $k=16$, and it is exactly
$98\%$ when $k=100$.

###  3.2 $k=4$

We have to contest these previous bounds: (15), (16), (17), (18), (19), (20).
Indeed, (21) is missing.

Corollary 2.6 beats all of these previous bounds when
$\frac{21}{4}<\lambda<\frac{13}{2}$. Hence it improves on the previous bounds
in over $31\%$ of the nontrivial range. Smaller choices of $t$ in Theorem 2.4
do not increase our range of improvement. While Theorem 2.3 part (ii) does
bring forth (21), it also fails to increase our range of improvement.

In the range $\lambda\leq\frac{21}{4}$, (17) is the strongest; we cannot even
match this. In the range $\lambda\geq\frac{13}{2}$, (18) is the strongest;
Theorem 2.3 part (i) recovers this.

###  3.3 $k=3$

In the case of cubes, Baier and Zhao proved the large sieve inequality [BZ05]

$\displaystyle\Delta_{3}(Q,N)\ll(Q^{4}+N^{\frac{9}{10}}Q^{\frac{6}{5}}+NQ^{\frac{6}{7}})Q^{\varepsilon}.$
(22)

This is the strongest bound in the range $\lambda\leq\frac{25}{7}$. Then we
see that (17) is the strongest in the range
$\frac{25}{7}\leq\lambda\leq\frac{9}{2}$, and (18) is the strongest in the
range $\lambda\geq\frac{9}{2}$. While we offer no improvement, Theorem 2.3
part (i) does match (18).

###  3.4 $k=2$

In the case of squares, Baier and Zhao gave the large sieve inequality [BZ08]

$\Delta_{2}(Q,N)\ll(Q^{3}+N+\min\\{NQ^{\frac{1}{2}}+N^{\frac{1}{2}}Q^{2}\\})Q^{\varepsilon}.$

This is the state of the art. While we offer no improvement, Theorem 2.3 part
(i) does match it when $\lambda\geq 3$.

##  4 From the large sieve to Farey fractions in short intervals

Let $Q,N$ and $k,s,t$ be positive integers. We also let $M$ be an integer, and
let $\\{z_{n}\\}$ be a sequence of complex numbers. The forthcoming implied
constants are independent of $Q,M,N$ and $\\{z_{n}\\}$. But they may depend on
$k,s,t$ or on an arbitrarily small positive real number $\varepsilon$. We
assume that

$\displaystyle Q^{k}\leq N\ll Q^{2k}.$

Dividing the summation over $q$ into dyadic intervals gives

$\displaystyle\sum_{q=1}^{Q}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}\ll
Q^{\varepsilon}\sup_{1\leq x\leq Q}\sum_{\frac{x}{2}<q\leq
x}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}.$

Let $x$ be a real number in the interval $1\leq x\leq Q$ and write

$\mathcal{A}_{k}(x):=\left\\{\frac{a}{q^{k}}:1\leq a\leq q^{k},\
\frac{x}{2}<q\leq x\ \mbox{and}\,\gcd(a,q)=1\right\\}$

so that

$\sum_{\frac{x}{2}<q\leq x}\sum_{\begin{subarray}{c}a=1\\\
\gcd(a,q)=1\end{subarray}}^{q^{k}}\left|\sum_{n=M+1}^{M+N}z_{n}e(na/q^{k})\right|^{2}=\sum_{\alpha\in\mathcal{A}_{k}(x)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}.$

We want to hit this with the general large sieve inequality (1), leading us to
consider the modulo 1 spacing of the set $\mathcal{A}_{k}(x)$. Let
$\delta=\delta(x)\leq x^{-k}$ be a positive real number. Note that
$\min\mathcal{A}_{k}(x)\geq x^{-k}$ and that $\max\mathcal{A}_{k}(x)\leq 1$.
Then, for every $\alpha,\beta\in\mathcal{A}_{k}(x)$, we have
$\lVert\alpha-\beta\rVert<\delta$ if and only if $|\alpha-\beta|<\delta$.
Hence, for every $\alpha_{0}\in\mathcal{A}_{k}(x)$, we have

$\mathcal{A}_{k}(x,\delta,\alpha_{0}):=\\{\alpha\in\mathcal{A}_{k}(x):\lVert\alpha-\alpha_{0}\rVert<\delta\\}=\\{\alpha\in\mathcal{A}_{k}(x):|\alpha-\alpha_{0}|<\delta\\}.$

Now observe that there exists a partition of the set $\mathcal{A}_{k}(x)$ into

$\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|$

parts so that each part is $\delta$-spaced modulo 1. Thus, for any one of
these parts $\mathcal{P}$, say, the general large sieve inequality (1) gives

$\sum_{\alpha\in\mathcal{P}}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq(N-1+\delta^{-1})\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$

This then yields the bound

$\sum_{\alpha\in\mathcal{A}_{k}(x)}\left|\sum_{n=M+1}^{M+N}z_{n}e(n\alpha)\right|^{2}\leq\left(\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\right)(N-1+\delta^{-1})\sum_{n=M+1}^{M+N}|z_{n}|^{2}.$

Putting everything together, we may take

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}\sup_{1\leq x\leq Q}\
\inf_{0<\delta\leq
x^{-k}}\left((N+\delta^{-1})\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\right).$

Recall that $x^{k}\leq Q^{k}\leq N$. Therefore
$\frac{1}{2N}\leq\frac{1}{2x^{k}}$, and so we may take

$\delta=\frac{1}{2N}$

in the infimum. Hence, with this choice of $\delta$, we have

$\displaystyle\Delta_{k}(Q,N)\ll Q^{\varepsilon}N\sup_{1\leq x\leq
Q}\sup_{\alpha_{0}\in\mathcal{A}_{k}(x)}|\mathcal{A}_{k}(x,\delta,\alpha_{0})|.$
(23)

##  5 From Farey fractions in short intervals to Waring’s problem

We keep the notation of the previous section. Recall that $x$ is a real number
in the interval $1\leq x\leq Q$ and that we have

$\delta=\frac{1}{2N}\leq\frac{1}{2x^{k}}.$

Let $\alpha_{0}\in\mathcal{A}_{k}(x)$. Then we write $a_{0}$ and $q_{0}$ for
the positive integers satisfying

$\alpha_{0}=\frac{a_{0}}{q_{0}^{k}},\qquad a_{0}\leq
q_{0}^{k},\qquad\frac{x}{2}<q_{0}\leq
x\qquad\mbox{and}\qquad\gcd(a_{0},q_{0})=1.$

Let $\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$ be the set of positive
integers $q$ in the dyadic interval $\frac{x}{2}<q\leq x$ for which there
exists an integer $b$ satisfying $|b|<\delta x^{2k}$ and $a_{0}q^{k}\equiv
b\bmod q_{0}^{k}$.

Given any $\alpha\in\mathcal{A}_{k}(x,\delta,\alpha_{0})$, we write
$\alpha=a/q^{k}$ in reduced form like we did for $\alpha_{0}$. Then we rewrite
the condition $|\alpha-\alpha_{0}|<\delta$ as $|a_{0}q^{k}-aq_{0}^{k}|<\delta
q^{k}q_{0}^{k}\leq\delta x^{2k}$. Therefore $\alpha\mapsto q$ defines a
function

$\mathcal{A}_{k}(x,\delta,\alpha_{0})\hookrightarrow\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0}).$

Since $\delta$ is sufficiently small, this function is injective. To see this,
take two fractions $\alpha,\beta\in\mathcal{A}_{k}(x,\delta,\alpha_{0})$ with
the same reduced denominator $q^{k}$. Now observe that
$|\alpha-\beta|\leq|\alpha-\alpha_{0}|+|\alpha_{0}-\beta|<2\delta$. Then we
see that $\alpha=\beta$, because otherwise we would get the contradictory
bound $|\alpha-\beta|\geq q^{-k}\geq x^{-k}\geq 2\delta$. Hence

$\displaystyle|\mathcal{A}_{k}(x,\delta,\alpha_{0})|\leq|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|.$
(24)

Now we let $\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$ be the set of
positive integers $q$ in the interval $s(\frac{x}{2})^{k}<q\leq sx^{k}$ for
which there exists an integer $b$ satisfying

$\displaystyle|b|<s\delta x^{2k}\qquad\mbox{and}\qquad a_{0}q\equiv b\bmod
q_{0}^{k}.$ (25)

Evidently, if
$q_{1},\ldots,q_{s}\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$, then
$q_{1}^{k}+\ldots+q_{s}^{k}\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$.
This leads us to consider, for each
$q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, the number
$R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)$ of $s$-tuples
$(q_{1},\ldots,q_{s})\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})^{s}$
satisfying $q=q_{1}^{k}+\ldots+q_{s}^{k}$. We have the formula

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}=\sum_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q).$

We proceed in two ways. First, we take the supremum to get

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}\leq|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\sup_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q).$
(26)

Second, we use Cauchy-Schwarz to get

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{2s}\leq|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\sum_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)^{2}.$
(27)

Either way, we need a bound on
$|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|$.

To count the number of $q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, we
first recall condition (25), from which we see that there are
$O(1+x^{2k}\delta)$ choices for $a_{0}q\bmod q_{0}^{k}$. Then, because
$\gcd(a_{0},q_{0}^{k})=1$, there are $O(1+x^{2k}\delta)$ choices for $q\bmod
q_{0}^{k}$. Note that $q\asymp x^{k}\asymp q_{0}^{k}$. Thus, for a fixed
residue $r$, there are $O(1)$ choices for $q$ with $q\equiv r\bmod q_{0}^{k}$.
Putting these estimates together, we have

$\displaystyle|\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})|\ll
1+x^{2k}\delta\ll N^{-1}Q^{2k}.$ (28)

###  5.1 Proof of Lemma 2.1

Here we continue with the first way.

For each $q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, we have
$R_{k,s}^{*}(x,\delta,a_{0},q_{0};q)\leq R_{k,s}(q)$. Furthermore, if
$q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})$, then $q\leq sx^{k}\leq
sQ^{k}$. Therefore

$\displaystyle\sup_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)\leq\sup_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R_{k,s}(q)\leq\sup_{q\leq
sQ^{k}}R_{k,s}(q).$

Combining this with bounds (24), (26) and (28), we have

$\displaystyle|\mathcal{A}_{k}(x,\delta,\alpha_{0})|^{s}\leq|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}\ll
N^{-1}Q^{2k}\sup_{q\leq sQ^{k}}R_{k,s}(q).$

We arrive at Lemma 2.1 by running this estimate through (23).

###  5.2 Proof of Lemma 2.2

Here we continue with the second way.

Note that

$\displaystyle\sum_{q\in\mathcal{B}^{*}_{k,s}(x,\delta,a_{0},q_{0})}R^{*}_{k,s}(x,\delta,a_{0},q_{0};q)^{2}$

counts the number of solutions to the equation

$\displaystyle p_{1}^{k}+\ldots+p_{s}^{k}=q_{1}^{k}+\ldots+q_{s}^{k}$ (29)

with each $p_{i},q_{i}\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$. By
orthogonality of exponentials, the integral

$\displaystyle\int_{0}^{1}\left|\sum_{n\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})}e(\alpha
n^{k})\right|^{2s}d\alpha$

also counts the number of solutions to (29).

Suppose that, uniformly in subsets $\mathcal{A}$ of positive integers $\leq
Q$, we have the bound

$\displaystyle\int_{0}^{1}\left|\sum_{n\in\mathcal{A}}e(\alpha
n^{k})\right|^{2s}d\alpha\ll D_{\varepsilon,k,s}(Q)\cdot|\mathcal{A}|^{s}.$

Recall that this is the supposition of Lemma 2.2. If
$q\in\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$, then $q\leq x\leq Q$. So this
mean value estimate holds with
$\mathcal{A}=\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})$. Combining this with
bounds (27) and (28), we have

$\displaystyle|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{2s}\ll
N^{-1}Q^{2k}\cdot
D_{\varepsilon,k,s}(Q)\cdot|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}.$

Hence, by bound (24), we have

$\displaystyle|\mathcal{A}_{k}(x,\delta,\alpha_{0})|^{s}\leq|\mathcal{A}^{*}_{k}(x,\delta,a_{0},q_{0})|^{s}\ll
N^{-1}Q^{2k}\cdot D_{\varepsilon,k,s}(Q).$

We arrive at Lemma 2.2 by running this through (23).

##  6 Proof of Theorem 2.3

Let $k$ and $n$ be positive integers. The following implied constants are
independent of $n$, but they may depend on $k$ or on an arbitrarily small
positive real number $\varepsilon$.

###  6.1 Part (i)

In this subsection, we assume that $k\geq 2$.

From the epsilonic power estimate on the divisor function, we have the bound

$\displaystyle R_{k,2}(n)\ll n^{\varepsilon}.$ (30)

###### Proof.

Assuming that $k$ is even, we have $R_{k,2}(n)\leq R_{2,2}(n)$. So, to finish
off this case, we merely apply the classical bound $R_{2,2}(n)\ll
n^{\varepsilon}$ [Gro85, Ch. 2, §. 4].

Assume that $k$ is odd, and suppose that $a$ and $b$ are positive integers
satisfying $a^{k}+b^{k}=n$. Recalling the factorisation
$a^{k}+b^{k}=(a+b)(a^{k-1}-a^{k-2}b+\cdots-ab^{k-2}+b^{k-1})$, we see that
$d=a+b$ divides $n$. Because $k>1$, we have $d<n<d^{k}$, and thus also
$0<d^{k}-n<n^{k}-n$. Computing $n=a^{k}+b^{k}\equiv d^{k}\bmod a$, we see that
$a$ divides $d^{k}-n$. Hence, as desired, we have

$R_{k,2}(n)\leq\sum_{\begin{subarray}{c}d\mid n\\\
d<n<d^{k}\end{subarray}}\sum_{a\mid(d^{k}-n)}1\ll\sum_{\begin{subarray}{c}d\mid
n\\\ d<n<d^{k}\end{subarray}}n^{\varepsilon/2}\ll n^{\varepsilon}.\qed$

To get Theorem 2.3 part (i), we merely insert estimate (30) into Lemma 2.1
with $s=2$.

###  6.2 Part (ii)

In this subsection, we assume that $k\geq 3$.

Using arithmetic algebraic geometry, Marmon derived the bound [Mar10, Thm.
1.3]

$\displaystyle R_{k,4}(n)\ll n^{\frac{1}{k}+\frac{2}{k\sqrt{k}}}.$ (31)

The proof is non-elementary, and we don’t copy it here. As suggested by Marc
Munsch, we get Theorem 2.3 part (ii) by inserting estimate (31) into Lemma 2.1
with $s=4$.

##  7 Proof of Theorem 2.4

Let $Q$ and $k,s,t$ be positive integers. Let $\mathcal{A}$ be a subset of
$\\{\text{positive integers $\leq Q$}\\}$. The forthcoming implied constants
are independent of $Q$ and $\mathcal{A}$, but they may depend on $k,s,t$ or on
an arbitrarily small positive real number $\varepsilon$.

From Wooley’s recent mean value estimate [Woo18, Thm. 1.1], we get the
following family of mean value estimates. For each $t\leq k$, we have

$\displaystyle\int_{0}^{1}\left|\sum_{n\in\mathcal{A}}e(\alpha
n^{k})\right|^{t(t+1)}d\alpha\ll
Q^{\frac{1}{2}t(t-1)+\varepsilon}|\mathcal{A}|^{\frac{1}{2}t(t+1)}.$ (32)

###### Proof.

We assume that $t\leq k$, and we put $s=\frac{1}{2}t(t+1)$. By orthogonality
of exponentials, the integral on the left-hand side of (32) counts the number
of solutions to the equation

$\displaystyle m_{1}^{k}+\ldots+m_{s}^{k}=n_{1}^{k}+\ldots+n_{s}^{k}$ (33)

with each $m_{i},n_{i}\in\mathcal{A}$. We now inflate this equation to a
partial Vinogradov system, a standard technique in the context of Waring’s
problem. Namely, we consider the system of $t$ equations

$\displaystyle m_{1}^{k}+\ldots+m_{s}^{k}$
$\displaystyle=n_{1}^{k}+\ldots+n_{s}^{k}$ (34) $\displaystyle
m_{1}^{t-1}+\ldots+m_{s}^{t-1}$ $\displaystyle=n_{1}^{t-1}+\ldots+n_{s}^{t-1}$
$\displaystyle\vdots$ $\displaystyle m_{1}+\ldots+m_{s}$
$\displaystyle=n_{1}+\ldots+n_{s}$

with each $m_{i},n_{i}\in\mathcal{A}$. As we explain in Subsection 7.1, when
we pass from counting solutions of equation (33) to counting solutions of this
partial Vinogradov system (34), we incur an inflationary factor of

$\displaystyle\ll\prod_{j=1}^{t-1}Q^{j}=Q^{\frac{1}{2}t(t-1)}.$

The number of solutions to our partial Vinogradov system is equal to

$\displaystyle\int_{(0,1]^{t}}\left|\sum_{n\in\mathcal{A}}e(\alpha_{1}n+\cdots+\alpha_{t-1}n^{t-1}+\alpha_{t}n^{k})\right|^{2s}d\alpha.$

So it suffices to show that this last integral is $\ll
Q^{\varepsilon}|\mathcal{A}|^{s}$, for which we apply [Woo18, Thm. 1.1]. ∎

To get Theorem 2.4, we merely insert estimate (32) into Lemma 2.2 with
$s=\frac{1}{2}t(t+1)$.

###  7.1 Explaining the inflationary factor

Here we introduce some quantities that depend on $\mathcal{A}$. Crucially, the
implied constants are independent of $\mathcal{A}$. For each positive integer
$\ell$, let $\mathcal{R}_{k,s}(\mathcal{A},\ell)$ be the set of $s$-tuples
$(n_{1},\ldots,n_{s})\in\mathcal{A}^{s}$ satisfying
$n_{1}^{k}+\ldots+n_{s}^{k}=\ell$, and let
$R_{k,s}(\mathcal{A},\ell)=|\mathcal{R}_{k,s}(\mathcal{A},\ell)|$ be the
number of these. Let

$\mathcal{B}_{k,s,t}(\mathcal{A},\ell)=\\{(n_{1}+\ldots+n_{s},\ldots,n_{1}^{t-1}+\ldots+n_{s}^{t-1}):(n_{1},\ldots,n_{s})\in\mathcal{R}_{k,s}(\mathcal{A},\ell)\\}$

so that
$(n_{1},\ldots,n_{s})\mapsto(n_{1}+\ldots+n_{s},\ldots,n_{1}^{t-1}+\ldots+n_{s}^{t-1})$
defines a function

$\mathcal{R}_{k,s}(\mathcal{A},\ell)\to\mathcal{B}_{k,s,t}(\mathcal{A},\ell).$

As per usual, we need to consider the size of the fibres of this function. For
each $(\ell_{1},\ldots,\ell_{t-1})\in\mathcal{B}_{k,s,t}(\mathcal{A},\ell)$,
let $V_{k,s,t}(\mathcal{A},\ell;\ell_{1},\ldots,\ell_{t-1})$ be the size of
the fibre. Then we have the formula

$R_{k,s}(\mathcal{A},\ell)=\sum_{(\ell_{1},\ldots,\ell_{t-1})\in\mathcal{B}_{k,s,t}(\mathcal{A},\ell)}V_{k,s,t}(\mathcal{A},\ell;\ell_{1},\ldots,\ell_{t-1}).$

By Cauchy-Schwarz, we have

$R_{k,s}(\mathcal{A},\ell)^{2}\leq|\mathcal{B}_{k,s,t}(\mathcal{A},\ell)|\sum_{(\ell_{1},\ldots,\ell_{t-1})\in\mathcal{B}_{k,s,t}(\mathcal{A},\ell)}V_{k,s,t}(\mathcal{A},\ell;\ell_{1},\ldots,\ell_{t-1})^{2}.$

Now we use the assumption that $\mathcal{A}\subseteq\\{\text{positive integers
$\leq Q$}\\}$ to get a bound on $|\mathcal{B}_{k,s,t}(\mathcal{A},\ell)|$.
This gives the inflationary factor. Indeed,

$|\mathcal{B}_{k,s,t}(\mathcal{A},\ell)|\ll\prod_{j=1}^{t-1}Q^{j}=Q^{\frac{1}{2}t(t-1)}.$

So we have

$R_{k,s}(\mathcal{A},\ell)^{2}\ll
Q^{\frac{1}{2}t(t-1)}\sum_{(\ell_{1},\ldots,\ell_{t-1})\in\mathcal{B}_{k,s,t}(\mathcal{A},\ell)}V_{k,s,t}(\mathcal{A},\ell;\ell_{1},\ldots,\ell_{t-1})^{2}.$

Now we sum over all $\ell$ of the form $\ell=n_{1}^{k}+\ldots+n_{s}^{k}$ with
$n_{1},\ldots,n_{s}\in\mathcal{A}$. This gives

$\sum_{\ell}R_{k,s}(\mathcal{A},\ell)^{2}\ll
Q^{\frac{1}{2}t(t-1)}\cdot\sum_{\ell}\sum_{(\ell_{1},\ldots,\ell_{t-1})\in\mathcal{B}_{k,s,t}(\mathcal{A},\ell)}V_{k,s,t}(\mathcal{A},\ell;\ell_{1},\ldots,\ell_{t-1})^{2}.$

Finally, we observe what these sums count. The sum on the left-hand side
counts the number of solutions to equation (33). The sums on the right-hand
side count the number of solutions to our partial Vinogradov system (34).

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Stephan Baier, Department of Mathematics, Ramakrishna Mission Vivekananda
Educational Research Institute, G. T. Road, PO Belur Math, Howrah, West Bengal
711202, India

E-mail address<EMAIL_ADDRESS>
Sean B. Lynch, Department of Pure Mathematics, School of Mathematics and
Statistics, UNSW Sydney, Sydney NSW 2052, Australia

E-mail address<EMAIL_ADDRESS>