$F,\bm{c}/\overline{\mathbb{Q}}$, we have $\operatorname{disc}(F,\bm{c})\neq
0$ if and only if $V(F,\bm{c}\cdot\bm{x})$ is smooth of codimension $2$.

######  Remark D.4.

Here $\operatorname{disc}(F,\bm{c})$ is $\pm 1$-unique, absolutely
irreducible, and _bihomogeneous_ of degrees (in the notation of
[benoist2012degres]*§1.1)

$\displaystyle(\deg_{1},\deg_{2})$
$\displaystyle=\left(\frac{1}{2-0}\cdot\frac{2^{m}-2^{m}}{2-2}+\frac{1}{0-2}\cdot\frac{2^{m}-0^{m}}{2-0},\frac{3}{2-0}\cdot\frac{0^{m}-2^{m}}{0-2}+\frac{3}{0-2}\cdot\frac{0^{m}-0^{m}}{0-0}\right)$
$\displaystyle=((m-1)2^{m-2},3\cdot 2^{m-2}).$

(Here we use [benoist2012degres]*Théorème 1.3. Note that $\deg_{var}=3\cdot
2^{m-2}$ and $m\cdot\deg_{var}=3\cdot\deg_{1}+1\cdot\deg_{2}$.) Also,
$V(F,\bm{c}\cdot\bm{x})/\mathbb{F}_{p}$ is smooth of codimension $2$ if and
only if $p\nmid\operatorname{disc}(F,\bm{c})$.

######  Proposition D.5.

In $\mathbb{Z}[F_{ijk},c_{1}/c_{m},\dots,c_{m-1}/c_{m},c_{m}]$, the formal
identity

$\operatorname{disc}(F,\bm{c})=\pm\operatorname{disc}(F|_{x_{m}=-(c_{1}x_{1}+\dots+c_{m-1}x_{m-1})/c_{m}})\cdot
c_{m}^{3\cdot 2^{m-2}}$

holds. In particular, both sides actually lie in $\mathbb{Z}[F_{ijk},\bm{c}]$.

###### Proof.

Call the left and right sides $L,R$, respectively (ignoring the sign, for
$R$). Let $\mathbb{A}\colonequals\operatorname{Spec}(k[F_{ijk},\bm{c}])$ over
$k\colonequals\overline{\mathbb{Q}}$, and let $U\colonequals
D_{\mathbb{A}}(c_{m})$. Then the “vanishing sets” $\lvert
V_{U}(L)\rvert,\lvert V_{U}(R)\rvert$ (not schemes) must coincide, by
definition of $L,R$.

So if we choose $d\geq 0$ with $c_{m}^{d}R\in k[\mathbb{A}]$, then $\lvert
V_{\mathbb{A}}(c_{m}^{d}R)\rvert\subseteq\lvert V_{\mathbb{A}}(c_{m}\cdot
L)\rvert$. Since $c_{m},L$ are _prime_ in $k[\mathbb{A}]$, it follows (by
Hilbert’s nullstellensatz) that $c_{m}^{d}R=\alpha\cdot c_{m}^{e}\cdot L^{f}$
for some “unit” $\alpha\in k[\mathbb{A}]^{\times}=k^{\times}$ (i.e. regular
function with $V_{\mathbb{A}}(\alpha)=\emptyset$) and exponents $e,f\geq 0$.

But in the variables $F_{ijk}$, we know that each of $L,R$ is homogeneous of
degree $(m-1)2^{m-2}$. So $f=1$. Yet in $c_{1},\dots,c_{m}$, each of $L,R$ is
homogeneous of degree $3\cdot 2^{m-2}$. Thus $e=d$, and $R=\alpha L$. In
particular, $R\in\mathbb{Z}[F_{ijk},\bm{c}/c_{m},c_{m}]\cap
k[F_{ijk},\bm{c}]=\mathbb{Z}[F_{ijk},\bm{c}]$.

It remains to show that $\alpha=\pm 1$. But
$L,R\in\mathbb{Z}[F_{ijk},\bm{c}]$, and $L$ is primitive by definition, so it
suffices to show that $R$ is primitive. But for $G$ in $x_{1},\dots,x_{m-1}$,
we have $R|_{(F,c_{m})=(G,1)}=\operatorname{disc}(G)\in\mathbb{Z}[G_{ijk}]$,
which is primitive by definition. Thus $R$ is primitive, as desired. ∎

###  D.2. Specializing to generic diagonal $F$

######  Definition D.6.

Given a _diagonal_ cubic form $F=F_{1}x_{1}^{3}+\dots+F_{m}x_{m}^{3}$ with
$m\geq 3$, let

$\Delta(F,\bm{c})\colonequals 3^{\mathfrak{e}_{m}}(F_{1}\cdots
F_{m})^{2^{m-2}}\prod_{\bm{\epsilon}}\left[\epsilon_{1}(c_{1}^{3}/F_{1})^{1/2}+\epsilon_{2}(c_{2}^{3}/F_{2})^{1/2}+\dots+\epsilon_{m}(c_{m}^{3}/F_{m})^{1/2}\right],$

where $\bm{\epsilon}$ ranges over $\\{1\\}\times\\{\pm 1\\}^{m-1}$. Here the
right side represents an element of $\mathbb{Z}[F_{i},\bm{c}]$, factored in
$\mathbb{Z}[F_{i},\bm{c}][(c_{i}^{3}/F_{i})^{1/2}]$.

######  Observation D.7.

$\Delta(F,\bm{c})$ is irreducible in $\overline{\mathbb{Q}}[F_{i},\bm{c}]$.

###### Proof.

Let $k\colonequals\overline{\mathbb{Q}}$ and $A\colonequals k[\bm{F},\bm{c}]$.
Consider “the” multi-quadratic ring extension $B\colonequals
A[\bm{E},\bm{d}]/(E_{i}^{2}-F_{i},d_{i}^{2}-c_{i})\cong_{k}k[\bm{E},\bm{d}]$.
Here $A,B$ are unique factorization domains, and $B$ is “Galois” over $A$, so
it suffices to show that the element

$P\colonequals E_{1}\cdots
E_{m}\left[\epsilon_{1}d_{1}^{3}/E_{1}+\dots+\epsilon_{m}d_{m}^{3}/E_{m}\right]=\epsilon_{1}d_{1}^{3}E_{2}\cdots
E_{m}+\dots+\epsilon_{m}d_{m}^{3}E_{1}\cdots E_{m-1}\in B$

is irreducible in $B$ for each fixed $\bm{\epsilon}$. (Note that
$\Delta(F,\bm{c})/3^{\mathfrak{e}_{m}}$ is a product of such $P$’s.)

But $P$ is _linear_ in $E_{m}$, with constant term $\pm d_{m}^{3}E_{1}\cdots
E_{m-1}$ coprime to the “slope” $d_{1}^{3}E_{2}\cdots E_{m-1}\pm\dots\pm
d_{m-1}^{3}E_{1}\cdots E_{m-2}$. So by Gauss’ lemma, $P$ is irreducible in
$B$. ∎

######  Remark D.8.

One might try to prove the observation by specializing $\bm{F}$ and using
Proposition-Definition 1.8, but this would seem tedious to fully work out
(since specialization can decrease degrees). On the other hand, since all
$m$-variable diagonal cubic forms over $\overline{\mathbb{Q}}$ are isomorphic,
one might try to “specialize” the observation to prove the irreducibility
statement in Proposition-Definition 1.8, but this is not a priori easy (since
$\overline{\mathbb{Q}}$ is not Hilbertian).

Nonetheless, we have _directly_ proven both the observation, and the
irreducibility statement in Proposition-Definition 1.8, using the same Galois-
theoretic ideas.

######  Corollary D.9.

If $F$ denotes a _generic diagonal_ cubic form
$F_{1}x_{1}^{3}+\dots+F_{m}x_{m}^{3}$, then the equality
$\operatorname{disc}(F,\bm{c})=\pm\Delta(F,\bm{c})$ holds in
$\mathbb{Z}[F_{i},\bm{c}]$.

###### Proof.

At the level of $\overline{\mathbb{Q}}$-points, the polynomials
$\operatorname{disc}(F,\bm{c}),\Delta(F,\bm{c})\in\mathbb{Q}[F_{i},\bm{c}]$
have the same zero locus,
$\\{(F,\bm{c}):V(F,\bm{c}\cdot\bm{x})\;\textnormal{is not smooth of
codimension $2$}\\}$. (For proof, use the Jacobian criterion to compute the
singularities of the projective system of equations $F=\bm{c}\cdot\bm{x}=0$.
The case $F_{1}\cdots F_{m}=0$ requires some care; one must prove
$\Delta(F,\bm{c})=0$ when $F_{i}=c_{i}=0$ for some $i\in[m]$, and also when
$F_{i}=F_{j}=0$ for some $\\{i,j\\}\in\binom{[m]}{2}$.)

Also, each side has total degree $(m+2)2^{m-2}$, and $\Delta(F,\bm{c})$ is
prime in $\mathbb{Q}[F_{i},\bm{c}]$. Hence the two sides are
$\mathbb{Q}$-proportional. To finish, note that equality holds when
$(F,\bm{c})=(x_{1}^{3}+\dots+x_{m-1}^{3},e_{m})$: by the proposition,
$\operatorname{disc}(F,\bm{c})$ becomes $\operatorname{disc}(F|_{x_{m}=0})=\pm
3^{\mathfrak{e}_{m}}$, while by direct computation, $\pm\Delta(F,\bm{c})$
becomes “$\pm 3^{\mathfrak{e}_{m}}F_{m}^{2^{m-2}}(c_{m}^{3}/F_{m})^{2^{m-2}}$”
(i.e. $\pm 3^{\mathfrak{e}_{m}}$). ∎

######  Remark D.10.

_Thus_ , the “diagonally restricted” $\operatorname{disc}(F,\bm{c})$ is
absolutely irreducible over $\mathbb{Q}$.

##  Appendix E An example of a desirable but non-standard conjecture

Throughout Appendix E, let $n\colonequals 3$, and let
$Q_{\bm{h}}\colonequals\sum_{i\in[n]}h_{i}y_{i}^{2}\in\mathbb{Z}[\bm{y}]$ for
each $\bm{h}\in\mathbb{Z}^{n}$. Also let $F_{0}(\bm{y})\colonequals
y_{1}^{3}+\dots+y_{3}^{3}$ and $F(\bm{x})\colonequals
x_{1}^{3}+\dots+x_{6}^{3}$.

###  E.1. A uniform conjecture on integer solutions to ternary quadratic
equations

######  Conjecture E.1.

_Uniformly_ over reals $X,H\geq 1$ and pairs
$(\bm{h},k)\in\mathbb{Z}^{3}\times\mathbb{Z}$ with
$\lVert\bm{h}\rVert\in[H,2H]$ and
$-h_{1}h_{2}h_{3}k\notin\\{0\\}\cup(\mathbb{Q}^{\times})^{2}$, we have

$\\#\\{\bm{y}\in\mathbb{Z}^{3}\cap[-X,X]^{3}:Q_{\bm{h}}(\bm{y})=k\\}\ll_{\epsilon}(XH)^{\epsilon}\cdot\left(\frac{X}{\lvert
h_{1}h_{2}h_{3}\rvert^{1/3}}+\left(X^{2}H\right)^{1/4}\right).$

######  Remark E.2.

In the positive-definite case $h_{1},h_{2},h_{3}>0$, the introduction of
[golubeva1998nonhomogeneous] mentions a similar “quite plausible” conjecture
asserting the $O(k^{1/4})$-boundedness of the $k$th coefficient of the cusp
form of weight $3/2$ corresponding to $Q_{\bm{h}}$, uniformly over $\bm{h}$.

######  Remark E.3.

Using [duke1993bounds, heath1996new]’s delta method along the lines of
[browning2017twisted], one can likely give a _heuristic_ argument for
Conjecture E.1, assuming “generic square-root cancellation” over the modulus
$q$ (for “generic” $\bm{c}$’s). Some details, including the analysis of the
oscillatory integral for small $q$’s, would likely require some care to flesh
out (cf. [browning2017twisted]), because $k\neq 0$.

(The “homogeneous case” $k=0$ would be easier to analyze—at least under
standard conjectures—but we have excluded it from Conjecture E.1 for
simplicity.)

######  Remark E.4.

The $X/\lvert h_{1}h_{2}h_{3}\rvert^{1/3}$ comes from a loose real-density
bound: note that

$\displaystyle(2\epsilon)^{-1}\operatorname{vol}\left(\bm{y}\in\prod_{i\in[3]}[X_{i},2X_{i}]:Q_{\bm{h}}(\bm{y})\in[k-\epsilon,k+\epsilon]\right)$
$\displaystyle\ll\frac{X_{1}X_{2}X_{3}}{\max_{i\in[3]}(\lvert h_{i}\rvert
X_{i}^{2})}\leq\frac{X_{1}X_{2}X_{3}}{\prod_{i\in[3]}(\lvert h_{i}\rvert
X_{i}^{2})^{1/3}}=\frac{(X_{1}X_{2}X_{3})^{1/3}}{\lvert
h_{1}h_{2}h_{3}\rvert^{1/3}}\ll\frac{X}{\lvert h_{1}h_{2}h_{3}\rvert^{1/3}}$

holds uniformly over
$(\bm{X},k,\epsilon)\in(0,X]^{3}\times\mathbb{R}\times(0,1]$, say. (For
“lopsided” $\bm{h}$, the bound $X/\lvert h_{1}h_{2}h_{3}\rvert^{1/3}$ can
likely be improved at the cost of cleanliness.)

######  Remark E.5.

Fix $(\bm{h},k)\in\mathbb{Z}^{3}\times\mathbb{Z}$ with $h_{1}+h_{2}+h_{3}=0$
and $k=-3F_{0}(\bm{h})$. Then $k=-9h_{1}h_{2}h_{3}$, so
$-h_{1}h_{2}h_{3}k=(3h_{1}h_{2}h_{3})^{2}$ is a square. And in fact, here
$Q_{\bm{h}}(\bm{y})=k$ does have many “trivial” solutions
$\bm{y}\in\mathbb{Z}^{3}$, including $(t+3h_{2},t-3h_{1},t)$ for each
$t\in\mathbb{Z}$. So in the range $X^{\delta}\leq\lVert\bm{h}\rVert\leq\delta
X$, for instance, $Q_{\bm{h}}(\bm{y})=k$ would have $\gg X$ solutions
$\bm{y}\in[-X,X]^{3}$ as $X\to\infty$.

Thus in Conjecture E.1, we need _some_ nontrivial requirement on $(\bm{h},k)$,
even if $h_{1}h_{2}h_{3}\neq 0$. Conjecture E.1’s assumption
“$-h_{1}h_{2}h_{3}k\notin\\{0\\}\cup(\mathbb{Q}^{\times})^{2}$” is thus quite
natural.

###  E.2. A cubic application via differencing

To bound $N_{F}(X)$, we will first “dilute and fiber $N_{F}(X)$ into ternary
affine quadrics” using multidimensional van der Corput differencing (inspired
by [marmon2019hasse]*Lemma 3.3, which similarly reduced a quartic problem—in
some ranges—to a problem about mixed cubic-quartic exponential sums) and then
apply Conjecture E.1 to bound most of the resulting fibers.

######  Remark E.6.

The basic idea of studying “quadratic fibers” of cubic Diophantine problems
also appears in [golubeva1998nonhomogeneous, wooley2013waring], for instance,
and dates back at least to [linnik1943representation]’s proof that $G(3)\leq
7$. (For more details on $G(3)$ and its history, see [vaughan2002waring].)

Since Conjecture E.1 does not apply to all
$(\bm{h},k)\in\mathbb{Z}^{3}\times\mathbb{Z}$, we will also need the following
conjecture:

######  Conjecture E.7.

Uniformly over reals $H>0$, we have

$\\#\\{(\bm{h},z)\in(\mathbb{Z}^{3}\setminus\\{\bm{0}\\})\times\mathbb{Z}:\lVert\bm{h}\rVert\leq
H\;\textnormal{and}\;3h_{1}h_{2}h_{3}(h_{1}^{3}+h_{2}^{3}+h_{3}^{3})=z^{2}\\}\ll_{\epsilon}H^{2+\epsilon}.$

######  Remark E.8.

For fixed $h_{1},h_{2}$ with $h_{1}h_{2}(h_{1}^{3}+h_{2}^{3})\neq 0$, the
equation $z^{2}=3h_{1}h_{2}h_{3}F_{0}(\bm{h})$ in $h_{3},z$ cuts out a genus
$1$ affine plane curve, which contains the points $(0,0)$ and
$(-h_{1}-h_{2},\pm 3h_{1}h_{2}h_{3})$—and thus defines an elliptic (in fact, a
Mordell) curve, namely

$y^{2}=3h_{1}h_{2}\left((h_{1}^{3}+h_{2}^{3})x^{3}+1\right),\quad\textnormal{where}\;(x,y)\colonequals(1/h_{3},z/h_{3}^{2}).$

In particular, we may view $\\{z^{2}=3h_{1}h_{2}h_{3}F_{0}(\bm{h})\\}$ as an
“elliptic fibration” over $h_{1},h_{2}$. Conjecture E.7 would thus follow from
a certain standard hypothesis on the rank growth of elliptic curves over
$\mathbb{Q}$ (which would itself follow from BSD and GRH, by
[iwaniec2004analytic]*Proposition 5.21); cf. [bonolis2020uniform]*discussion
after Theorem 1.1.

What can be said unconditionally towards Conjecture E.7? (The fact that we are
interested only in certain integral points, rather than all rational points,
may help.)

######  Proposition E.9.

Assume Conjectures E.1 and E.7. Then
$N_{F}(X)\ll_{\epsilon}X^{45/13+\epsilon}$ as $X\to\infty$.

######  Remark E.10.

$45/13=3.4615\ldots<7/2$.

###### Proof.

Fix a small absolute constant $c>0$ to be specified later. Then fix a nonzero
nonnegative weight $\nu\in C^{\infty}_{c}(\mathbb{R}^{n})$ supported on
$[1,1+c]^{n}$. (Recall that $n\colonequals 3$.)

Now let $w(\bm{y},\bm{z})\colonequals\nu(\bm{y})\nu(-\bm{z})$. Then by Hölder
and dyadic decomposition, it certainly suffices to show that
$N_{F,w}(X)\ll_{\epsilon,\nu}X^{45/13+\epsilon}$. But

$N_{F,w}(X)=\sum_{a\in\mathbb{Z}}N_{F_{a},\nu}(X)^{2}.$

Now fix a van der Corput differencing set
$\mathcal{H}\subseteq\mathbb{Z}^{n}\cap[0,X]^{n}$, let
$\nu_{X}\colonequals\nu\circ X^{-1}$ and $w_{X}\colonequals w\circ X^{-1}$ for
convenience, and “dilute” $N_{F_{a},\nu}(X)$ by $\mathcal{H}$ to get

$N_{F_{a},\nu}(X)\colonequals\sum_{\bm{x}_{0}\in\mathbb{Z}^{n}}\nu_{X}(\bm{x}_{0})\bm{1}_{F_{0}(\bm{x}_{0})=a}=\sum_{\bm{x}_{0}\in\mathbb{Z}^{n}}\lvert\mathcal{H}\rvert^{-1}\sum_{\bm{h}\in\mathcal{H}}\nu_{X}(\bm{x})\bm{1}_{F_{0}(\bm{x})=a},$

where $\bm{x}\colonequals\bm{x}_{0}+\bm{h}$. Then by Cauchy over
$\bm{x}_{0}\ll X$, it follows uniformly over $a\in\mathbb{Z}$ that

$\displaystyle N_{F_{a},\nu}(X)^{2}$ $\displaystyle\ll
X^{n}\sum_{\bm{x}_{0}\in\mathbb{Z}^{n}}\lvert\mathcal{H}\rvert^{-2}\sum_{\bm{h}_{1},\bm{h}_{2}\in\mathcal{H}}\nu_{X}(\bm{x}_{2})\bm{1}_{F_{0}(\bm{x}_{2})=a}\cdot\nu_{X}(\bm{x}_{1})\bm{1}_{F_{0}(\bm{x}_{1})=a}$
$\displaystyle=X^{n}\lvert\mathcal{H}\rvert^{-2}\sum_{\bm{x}_{1},\bm{x}_{2}\in\mathbb{Z}^{n}}D(\bm{x}_{2}-\bm{x}_{1})\cdot\nu_{X}(\bm{x}_{2})\bm{1}_{F_{0}(\bm{x}_{2})=a}\cdot\nu_{X}(\bm{x}_{1})\bm{1}_{F_{0}(\bm{x}_{1})=a}$
$\displaystyle=X^{n}\lvert\mathcal{H}\rvert^{-2}\sum_{\bm{h}\in\mathbb{Z}^{n}}D(\bm{h})\sum_{\bm{x}_{1}\in\mathbb{Z}^{n}}\nu_{X,\bm{h}}(\bm{x}_{1})\bm{1}_{F_{0}(\bm{x}_{1}+\bm{h})=F_{0}(\bm{x}_{1})=a},$

where
$D(\bm{h})\colonequals\\#\\{(\bm{h}_{1},\bm{h}_{2})\in\mathcal{H}^{2}:\bm{h}=\bm{h}_{2}-\bm{h}_{1}\\}\leq\\#\mathcal{H}$
and $\nu_{X,\bm{h}}(\bm{y})\colonequals\nu_{X}(\bm{y}+\bm{h})\nu_{X}(\bm{y})$.
Summing over $a\in\mathbb{Z}$, and letting $\bm{y}\colonequals\bm{x}_{1}$,
leads to the bound

$\displaystyle N_{F,w}(X)$ $\displaystyle\ll
X^{n}\lvert\mathcal{H}\rvert^{-2}\sum_{\bm{h}\in\mathbb{Z}^{n}}D(\bm{h})\sum_{\bm{y}\in\mathbb{Z}^{n}}\nu_{X,\bm{h}}(\bm{y})\bm{1}_{F_{0}(\bm{y}+\bm{h})=F_{0}(\bm{y})}$
$\displaystyle\ll
X^{n}\lvert\mathcal{H}\rvert^{-1}\sum_{\bm{h}\in\mathcal{H}-\mathcal{H}}\sum_{\bm{y}\in\mathbb{Z}^{n}}\bm{1}_{\bm{y}+\bm{h},\bm{y}\in[X,(1+c)X]^{n}}\bm{1}_{F_{\bm{h}}(\bm{y})=0},$

where $F_{\bm{h}}(\bm{y})\colonequals F_{0}(\bm{y}+\bm{h})-F_{0}(\bm{y})$.

Inspired by [marmon2019hasse], we now let $K\colonequals cX^{\theta}$ for an
exponent $\theta\in(0,1]$ to be specified, and let
$\mathcal{H}\colonequals\\{\bm{d}\in[0,cX]\times[0,cK]^{n-1}:6\mid\bm{d}\\}$.
The key point is that _if_
$(\bm{h},\bm{y})\in(\mathcal{H}-\mathcal{H})\times[X,(1+c)X]^{n}$ and $\lvert
h_{1}\rvert\geq K$, then $\lvert h_{2}\rvert,\lvert h_{3}\rvert\leq cK\leq
c\lvert h_{1}\rvert$, so the mean value theorem for $y\mapsto y^{3}$ implies

$\lvert F_{\bm{h}}(\bm{y})\rvert\geq\lvert h_{1}\rvert\cdot(3-o_{c\to
0}(1))X^{2}-(\lvert h_{2}\rvert+\lvert h_{3}\rvert)\cdot(3+o_{c\to
0}(1))X^{2}\geq\lvert h_{1}\rvert X^{2}$

(provided we chose $c$ to be sufficiently small), whence
$F_{\bm{h}}(\bm{y})\neq 0$. Thus

$N_{F,w}(X)\ll
X^{n}\lvert\mathcal{H}\rvert^{-1}\sum_{\bm{h}\in\mathbb{Z}^{n}}\bm{1}_{\bm{h}\in[-K,K]^{n}}\bm{1}_{6\mid\bm{h}}\sum_{\bm{y}\in\mathbb{Z}^{n}}\bm{1}_{\bm{y}\in[X,(1+c)X]^{n}}\bm{1}_{F_{\bm{h}}(\bm{y})=0}.$

For each $\bm{h},\bm{y}$ contributing to the sum above, let
$\bm{h}^{\prime}\colonequals\bm{h}/6\in\mathbb{Z}^{n}\cap[-K/6,K/6]^{n}$ and
$\bm{y}^{\prime}\colonequals\bm{y}+3\bm{h}^{\prime}\in\mathbb{Z}^{n}\cap[(1-c)X,(1+2c)X]^{n}$;
then

$F_{\bm{h}}(\bm{y})=\sum_{i\in[n]}[(y_{i}+h_{i})^{3}-y_{i}^{3}]=\sum_{i\in[n]}[3h_{i}(y_{i}+h_{i}/2)^{2}+h_{i}^{3}/4]=18Q_{\bm{h}^{\prime}}(\bm{y}^{\prime})+54F_{0}(\bm{h}^{\prime}).$

Let $k_{\bm{h}^{\prime}}\colonequals-3F_{0}(\bm{h}^{\prime})$; then by
Conjectures E.7 and E.1, and the “trivial bound”

$\displaystyle\\#\\{\bm{y}^{\prime}\in\mathbb{Z}^{n}\cap[-2X,2X]^{n}:Q_{\bm{h}^{\prime}}(\bm{y}^{\prime})=k_{\bm{h}^{\prime}}\\}$
$\displaystyle\ll
X^{n}\bm{1}_{\bm{h}^{\prime}=\bm{0}}+X^{n-1}\left(\sum_{\pi\in
S_{3}}\bm{1}_{h^{\prime}_{\pi(1)}=h^{\prime}_{\pi(2)}=0}+\sum_{\pi\in
S_{3}}\bm{1}_{h^{\prime}_{\pi(1)}=h^{\prime}_{\pi(2)}+h^{\prime}_{\pi(3)}=0}\right)+O_{\epsilon}(X^{n-2+\epsilon})$

(proven later, soon below), we conclude that

$\displaystyle N_{F,w}(X)$
$\displaystyle\ll_{c,\epsilon}\frac{X^{n}(XK)^{\epsilon}}{1+XK^{n-1}}\left(X^{n}+KX^{n-1}+K^{2}X^{n-2}+\sideset{}{{}^{\prime}}{\sum}_{\bm{h}^{\prime}\ll
K}\left(\frac{X}{\lvert
h^{\prime}_{1}h^{\prime}_{2}h^{\prime}_{3}\rvert^{1/3}}+\left(X^{2}K\right)^{1/4}\right)\right)$
$\displaystyle\ll_{c,\epsilon}\frac{X^{n+2\epsilon}}{1+XK^{n-1}}\left(X^{n}+KX^{n-1}+K^{2}X^{n-2}+K^{2n/3}X+K^{n}\left(X^{2}K\right)^{1/4}\right),$

where the sum over $\bm{h}^{\prime}$ is restricted to
$\\{-h^{\prime}_{1}h^{\prime}_{2}h^{\prime}_{3}k_{\bm{h}^{\prime}}\notin\\{0\\}\cup(\mathbb{Q}^{\times})^{2}\\}$.
Finally, recall that $n\colonequals 3$, and set
$\theta\colonequals(4/13)\cdot(5/2)=10/13$ to obtain the desired bound of
$X^{45/13+\epsilon}$ as $X\to\infty$. ∎

###### Loose ends.

To derive the “trivial bound” for $\\#\\{\bm{y}^{\prime}\ll
X:Q_{\bm{h}^{\prime}}(\bm{y}^{\prime})=k_{\bm{h}^{\prime}}\\}$, note that

  1. (1)

uniformly over $(a,b,t)\in(\mathbb{Z}\setminus\\{0\\})^{2}\times\mathbb{Z}$,
we have

$\\#\\{(u,v)\in[-2X,2X]^{2}:au^{2}+bv^{2}=t\\}\ll_{\epsilon}X\cdot\bm{1}_{t=0}+(\lvert
ab\rvert X)^{\epsilon},$

by using $d(t)\cdot\bm{1}_{t\neq 0}\ll_{\epsilon}\lvert t\rvert^{\epsilon}$ if
$-ab$ is a square, and by working in $\mathbb{Q}(\sqrt{-ab})$ otherwise
(noting that $p^{e}$ has at most $(e+1)^{2}=d(p^{e})^{2}$ ideal divisors in
$\mathbb{Q}(\sqrt{-ab})$, and that if $-ab>0$ then the fundamental unit
$\varepsilon$ is $\geq\min(1+\sqrt{2},\frac{1+\sqrt{5}}{2})$);

  2. (2)

if $h^{\prime}_{1}h^{\prime}_{2}h^{\prime}_{3}\neq 0$, then
$k_{\bm{h}^{\prime}}-h^{\prime}_{3}(y^{\prime}_{3})^{2}$ is _nonzero_ for all
but at most $O(1)$ integers $y^{\prime}_{3}$;

  3. (3)

if $\pi\in S_{3}$ and $h^{\prime}_{\pi(1)}=0$, but
$h^{\prime}_{\pi(2)}h^{\prime}_{\pi(3)}(h^{\prime}_{\pi(2)}+h^{\prime}_{\pi(3)})\neq
0$, then $h^{\prime}_{\pi(2)}h^{\prime}_{\pi(3)}k_{\bm{h}^{\prime}}\neq 0$;
and

  4. (4)

if $\pi\in S_{3}$ and
$h^{\prime}_{\pi(1)}=h^{\prime}_{\pi(2)}(h^{\prime}_{\pi(2)}+h^{\prime}_{\pi(3)})=0$,
but $\bm{h}^{\prime}\neq\bm{0}$, then $h^{\prime}_{\pi(3)}\neq 0$, whence for
each $y^{\prime}_{\pi(1)},y^{\prime}_{\pi(2)}$ there are $\leq O(1)$ integers
$y^{\prime}_{\pi(3)}$ with
$Q_{\bm{h}^{\prime}}(\bm{y}^{\prime})=k_{\bm{h}^{\prime}}$.

The desired “trivial bound” follows upon combining (1)–(4) in a fairly simple
manner. ∎

###  E.3. Commentary on the proof of the proposition

######  Remark E.11.

One can certainly relax the assumptions of Proposition E.9. For example, an
“$\ell^{1}$-average” version of Conjecture E.1 (over $\bm{h}\ll K$) would
suffice in place of Conjecture E.1. Also, we could relax the exponent in
Conjecture E.7 from $2+\epsilon$ to $2.6+\epsilon$.

In a more qualitative direction, note that $y\mapsto y^{3}$ is increasing. So
if $\bm{h}\neq\bm{0}$, and $\bm{h}\geq 0$ (or $\bm{h}\leq 0$), then
$F_{\bm{h}}(\bm{y})>0$ (resp. $F_{\bm{h}}(\bm{y})<0$) holds for all
$\bm{y}\in\mathbb{R}^{n}$. Thus we only need Conjecture E.1 in the
_indefinite_ case (i.e. when $h_{1},h_{2},h_{3}$ do not all have the same
sign)—and in fact, we may further assume that
$\max(\bm{h})\asymp\lvert\min(\bm{h})\rvert$.

######  Remark E.12.

At the beginning, we localized to $x\in[X,(1+c)X]$. In fact, we could have
localized to $\lvert x-X\rvert\leq X/\log{X}$ or $\lvert x-X\rvert\leq
X^{1-\epsilon_{0}}$, say, but it is unclear if this would help.

######  Remark E.13.

For approximate lattices $\mathcal{H}$, the estimate
$D(\bm{h})\ll\\#\mathcal{H}$ is close to the truth.

######  Remark E.14.

The “lopsided” choice of $\mathcal{H}$ in Proposition E.9 saves a factor of
roughly $K/X$ over what we would have gotten from a more “uniform” choice of
$\mathcal{H}$. This is essentially the observation behind the “averaged van
der Corput differencing” in [marmon2019hasse]*§3.2—which perhaps originated in
work of Heath-Brown [heath2007cubic]*§4. The point is, in order for something
like $h_{1}x_{1}^{2}+h_{2}x_{2}^{2}+h_{3}x_{3}^{2}$ to vanish (or be small at
all), two of the $h_{i}x_{i}^{2}$ must be comparable. Above we only used a
real (archimedean) version of this observation; is there a useful way to also
introduce divisibility (non-archimedean) conditions on $\mathcal{H}$?

And is there a more conceptual justification for our choice of $\mathcal{H}$
above? Choosing $\mathcal{H}$ still seems to be an art. What is the “best”
choice of $\mathcal{H}$? It may be worth trying
$\mathcal{H}\approx[cX]^{2}\times[cK]$, for instance—or more generally,
$\mathcal{H}\approx[cX]\times[cK_{2}]\times[cK_{3}]$ with $X\gg K_{2}\gg
K_{3}$.

######  Remark E.15.

Instead of applying van der Corput differencing to $N_{F_{a},\nu}(X)$ (once
for each $a\in\mathbb{Z}$), we could have applied van der Corput differencing
to $T(\theta)^{n}=T(\theta)^{3}$ (once for each
$\theta\in\mathbb{R}/\mathbb{Z}$), where $T(\theta)$ denotes a weighted cubic
Weyl sum. (Recall that $N_{F}(X)$ is the sixth moment of a certain cubic Weyl
sum.) However, at least with the approach above, the end results do not seem
to differ.

######  Remark E.16.

If one wanted to “amplify” the Hua problem (for $n=3$) to some $n\geq 4$—and
perhaps replace Proposition E.9 with a question about $n$-variable quadratic
forms (instead of ternary ones)—one would need to “restrict to minor arcs”
(though the precise definition, including the initial choice of Dirichlet
covering of $\mathbb{R}/\mathbb{Z}$, might require a little care).

######  Remark E.17.

To the author’s knowledge, all known unconditional proofs of the Hua bound
$N_{F}(X)\ll_{\epsilon}X^{7/2+\epsilon}$ interpolate between (unconditional)
results in $L^{4}$ and $L^{p}$ for some $p\in[8,\infty]$. The simplest such
“ingredients” might be Hua’s lemma in $L^{4}$ and $L^{8}$, which can be proven
by applying Weyl differencing once and twice, respectively.777There are more
elaborate arguments for the fourth and eight moments (roughly due to geometry
and smooth numbers, respectively) that remove Hua’s $\epsilon$’s, or even give
log-power savings for $N_{F}(X)$ (coming from the $L^{8}$ minor arcs; see
[vaughan2020some] for details), but here we focus on classical differencing.

Thus in some sense, Hua used Weyl differencing “strictly between $1$ and $2$
times” for the sixth moment. And Proposition E.9 can be viewed as an attempt
to “perturb Hua” using a single (_more general_) van der Corput differencing
_directly applied_ to the sixth moment.

###  E.4. Final questions and remarks

######  Remark E.18.

[mahler1936note] used the identity
$(9u^{4})^{3}+(3uv^{3}-9u^{4})^{3}+(v^{4}-9u^{3}v)^{3}=v^{12}$ to show that
$r_{3,3}(N)\gg N^{1/12}$ for $N=v^{12}>0$. Although “localizing” to $x\approx
X$ would rule out such solutions (one would need $9u^{4}\approx
3uv^{3}-9u^{4}$, i.e. $v^{3}\approx 6u^{3}$, but then
$v^{4}-9u^{3}v\approx(6u^{3}-9u^{3})v<0$—a contradiction), this example
suggests that some bases or fibrations are more biased than others. How biased
is the “differenced” basis used in Proposition E.9?

######  Remark E.19.

Suppose one applied the delta method directly on $r_{3,3}(N)$ to see what one
gets assuming the most optimistic square-root cancellation over the modulus
$q$ (but no further cancellation over the dual variable $\bm{c}$). If $X\asymp
N^{1/3}$, then one should expect roughly
$r_{3,3}(N)\ll_{\epsilon}X^{3s/4-3/2+\epsilon}=X^{3/4+\epsilon}$, by
extrapolating [heath1998circle]’s work for $s=4,6$ to $s=3$.

The “total heuristic bound” $\sum_{\bm{y}\ll
X}r_{3}(F_{0}(\bm{y}))\ll_{\epsilon}X^{3.75+\epsilon}$ seems to arise
naturally in other _exact_ 888i.e. “dilution-free” (no nontrivial van der
Corput differencing), etc. “complete geometric exponential sum” approaches to
bounding $N_{F}(X)$ as well:

  1. (1)

If one applies Deligne rather than GRH in [hooley_greaves_harman_huxley_1997,
heath1998circle], one gets $N_{F}(X)\ll_{\epsilon}X^{3.75+\epsilon}$
(rigorously, in fact—but unfortunately, this is worse than the Hua bound).

  2. (2)

The delta method heuristic for Conjecture E.1 has an error term of
$(X^{2}H)^{1/4+\epsilon}$,999the heuristic being uniform in $k$, though
$k=-3F_{0}(\bm{h})$ is what is relevant to us which for $H\asymp X$ gives
$X^{3/4+\epsilon}$ for each $\bm{h}$.

Is the $X^{3/4+\epsilon}$ commonality a red herring? It might be interesting
to see if there is a deeper connection between the cubic hyperplane section
Hasse–Weil $L$-functions of [hooley1986HasseWeil, heath1998circle] on the one
hand, and the various Kloosterman–Salié sums and modular forms associated to
quadratic problems arising from Weyl or van der Corput differencing on the
other.

######  Remark E.20.

It may be interesting to try attacking other problems—including some of those
listed in Appendix B—by differencing or fibering.

##  Appendix F Using Jutila’s circle method instead of the delta method

In analytic number theory, it is sometimes easier to work with numbers in
_special_ , rather than general, sequences. For a nice survey of applications
of _smooth_ numbers in particular, see [granville2008smooth]*§6, which however
predates Zhang’s famous breakthrough work [zhang2014bounded] (surveyed in
[granville2015primes]; note that [zhang2014bounded]’s “reduction to smooth
moduli” was considered earlier in [motohashi2008smoothed], but in fact
discovered _independently_ by Zhang).

In particular, it is conceivable that the large sieves considered in
Definition 1.24 could be easier to approach for special (e.g. smooth) moduli
than for general moduli, so it makes sense to try to “assess” the possibility
of restricting to special moduli in the circle method. In general, “moduli
restriction” in the circle method—i.e. “Jutila’s circle method”—dates back to
[jutila1992transformations, jutila_greaves_harman_huxley_1997] (but see
[jutila2007distribution] for a more general and accessible reference).
Jutila’s circle method has not yet been applied to classical Diophantine
problems, but as far as the author can tell, there is no _known_ fundamental
barrier to doing so.

To highlight the relevant issues, we now focus on a concrete setting. For each
$X\geq 1$, let $T(\theta)\colonequals\sum_{\lvert x\rvert\leq X}e(\theta
x^{3})$ and $Y\colonequals X^{3/2}$. For simplicity, restrict attention to the
“most generic” minor arcs in the _traditional_ circle method—i.e. to
$\mathfrak{m}^{\prime}\colonequals\bigcup_{n\asymp
Y}\bigcup^{\prime}_{a\bmod{n}}\\{\lvert\theta-a/n\rvert\leq 1/(nY)\\}$, the
union of “standard Dirichlet arcs” to modulus $n\asymp Y$.

In the setting above, the first key question to answer seems to be the
following:

######  Question F.1.

For each $X\geq 1$, let $\mathcal{Q}=\mathcal{Q}(Y)$ be a “natural” dense set
of integers $q\asymp Y$, e.g. the set of $Y^{0.01}$-smooth integers $q\asymp
Y$. Let $L\colonequals\sum_{q\in\mathcal{Q}}\phi(q)$ (cf.
[jutila2007distribution]*(2.1)). Let $A>0$ be a parameter, growing slowly with
$X$, such that $AL/Y^{2}\to\infty$ as $X\to\infty$. What can one say about the
“covering efficiency ratio”

$\rho(\mathfrak{m}^{\prime},\mathcal{Q},A)\colonequals\frac{AL/Y^{2}}{\inf_{\theta\in\mathfrak{m}^{\prime}}\\#\\{q\asymp
Y\;\textnormal{and}\;b\in(\mathbb{Z}/q\mathbb{Z})^{\times}:\lVert\theta-b/q\rVert_{\mathbb{R}/\mathbb{Z}}\leq
A/Y^{2}\\}}$

as $X\to\infty$? Is it necessarily true that
$\rho(\mathfrak{m}^{\prime},\mathcal{Q},A)\ll_{\epsilon}X^{\epsilon}$ as
$X\to\infty$?

To our knowledge, this question has not been explored much, beyond some rather
general $\ell^{2}$ results such as [jutila2007distribution]*(2.4). We end the
present Appendix F with some remarks.

######  Remark F.2.

Fix a choice of $\mathcal{Q},A$. Say
$\rho(\mathfrak{m}^{\prime},\mathcal{Q},A)\ll_{\epsilon}X^{\epsilon}$. Then at
least for “totally positive” integrals, one can freely (up to a multiplicative
loss of $X^{\epsilon}$, and an “$A$-widening” of Dirichlet arcs) pass to the
corresponding “restricted” integrals. For instance,

$\displaystyle\int_{\theta\in\mathfrak{m}^{\prime}}d\theta\,\lvert
T(\theta)\rvert^{6}$
$\displaystyle\leq\frac{1}{\inf_{\theta\in\mathfrak{m}^{\prime}}\\#\cdots}\cdot\sum_{q\in\mathcal{Q}}\sideset{}{{}^{\prime}}{\sum}_{b\bmod{q}}\int_{\lvert\theta-b/q\rvert\leq
A/Y^{2}}d\theta\,\lvert T(\theta)\rvert^{6}$
$\displaystyle\ll_{\epsilon}\frac{Y^{2}}{L}\cdot\sum_{q\in\mathcal{Q}}\sideset{}{{}^{\prime}}{\sum}_{b\bmod{q}}\frac{X^{\epsilon}}{A}\int_{\lvert\theta-b/q\rvert\leq
A/Y^{2}}d\theta\,\lvert T(\theta)\rvert^{6}.$

Provided $A,Y^{2}/L\ll_{\epsilon}X^{\epsilon}$, one should then be able to
prove $\int_{\theta\in\mathfrak{m}^{\prime}}d\theta\,\lvert
T(\theta)\rvert^{6}\ll_{\epsilon}X^{3+\epsilon}$ under a large sieve (as in
Definition 1.24 but) _restricted_ to moduli $q\in\mathcal{Q}$.

######  Remark F.3.

The setup above fails to cover minor arcs “far away” from
$\mathfrak{m}^{\prime}$: if we fix reduced fractions $a/n,b/q$ with $n\ll
q\asymp Y$, where $n\notin\mathcal{Q}$ and $q\in\mathcal{Q}$, then $\lVert
a/n-b/q\rVert_{\mathbb{R}/\mathbb{Z}}\geq 1/(nq)\gg 1/(nY)$, so if $n\ll Y/A$
is sufficiently _small_ , then the point $a/n$ must lie _outside_ the arc
$\lVert\theta-b/q\rVert_{\mathbb{R}/\mathbb{Z}}\leq A/Y^{2}$. (Even if we
relax the restriction $q\asymp Y$ to $q\ll Y$, and replace $A/Y^{2}$ with
$A/(qY)$, a similar problem remains.)

So on such “non-extreme” minor arcs, a different setup would be needed.
Fortunately, such minor arcs seem to be simpler from other points of view. For
instance, the improved integral estimates in Paper III may help to simplify
matters for small $n$. Alternatively, as suggested in Appendix B, a partial
“reduction” from $6$ variables to $4$ variables may be possible for “non-
extreme” minor arcs.

It is also plausible that one could “efficiently include” (or otherwise
“efficiently handle”) such minor arcs by clever means that have eluded the
author.

## Acknowledgements

I thank my advisor, Peter Sarnak, for asking me to replace
[hooley1986HasseWeil, hooley_greaves_harman_huxley_1997, heath1998circle]’s
GRH assumption with a zero-density hypothesis (done in
[density_beats_hua_when])—and to “axiomatize” the Bombieri–Vinogradov zero-
density theorem (done in [axiomatize_bombieri-huxley])—which ultimately led to
the present paper.††This work was partially supported by NSF grant
DMS-1802211. I also thank him for many encouraging discussions, helpful
comments, and references to related works and problems, especially regarding
large sieve inequalities, general reciprocity conjectures, and zero-density
hypotheses.

Thanks also to Levent Alpoge and Will Sawin for fun, supportive, and
interesting discussions, especially at the early stages of
[density_beats_hua_when]; and to Will again for initial help with abelian
varieties (in the context of Proposition C.1). Finally—again regarding
[density_beats_hua_when]—thanks to Yang Liu, Evan O’Dorney, Ashwin Sah, and
Mark Sellke for conversations illuminating the combinatorics of an older
“counting version” of the present Lemma 2.14.

## References


# Emergent Cooperation for Energy-efficient Connectivity via Wireless Power
Transfer

Winston Hurst, Anurag Pallaprolu and Yasamin Mostofi Winston Hurst, Anurag
Pallaprolu, and Yasamin Mostofi are with the Department of Electrical and
Computer Engineering, University of California, Santa Barbara, USA (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>ymostofi@ece.ucsb.edu). This
work was supported in part by NSF RI award 2008449 and in part by ONR award
N00014-23-1-2715.

###### Abstract

This paper addresses the challenge of incentivizing energy-constrained, non-
cooperative user equipment (UE) to serve as cooperative relays. We consider a
source UE with a non-line-of-sight channel to an access point (AP), where
direct communication may be infeasible or may necessitate a substantial
transmit power. Other UEs in the vicinity are viewed as relay candidates, and
our aim is to enable energy-efficient connectivity for the source, while
accounting for the self-interested behavior and private channel state
information of these candidates, by allowing the source to “pay” the
candidates via wireless power transfer (WPT). We propose a cooperation-
inducing protocol, inspired by Myerson auction theory, which ensures that
candidates truthfully report power requirements while minimizing the expected
power used by the source. Through rigorous analysis, we establish the
regularity of valuations for lognormal fading channels, which allows for the
efficient determination of the optimal source transmit power. Extensive
simulation experiments, employing real-world communication and WPT parameters,
validate our theoretical framework. Our results demonstrate a 91% reduction in
outage probability with as few as 4 relay candidates, compared to the non-
cooperative scenario, and as much as 48% source power savings compared to a
baseline approach, highlighting the efficacy of our proposed methodology.

##  I Introduction

The forthcoming 6G wireless standard promises to revolutionize wireless
communication by delivering significant improvements in reliability, data
rates, and energy efficiency [1]. Achieving the desired technical performance
depends on a number of emerging technologies and paradigms, which in turn
bring their own challenges. For instance, high frequency communication (e.g.,
mmWave, THz) enables vastly increased capacity, but suffers from acute
shadowing and penetration losses [2]. At the same time, given the ever-
increasing number of connected devices, there is a growing need for scalable
system design, with reduced reliance on centralized coordination mechanisms,
to produce the desired performance.

This paper proposes the use of wireless power transfer (WPT) [3] to induce
cooperative behavior among non-cooperative communication devices, with the end
goal of enabling connectivity and/or reducing energy consumption. More
specifically, we consider a setting in which a large scale blockage (e.g., a
wall) obstructs the line-of-sight (LoS) link between a user equipment (UE) and
an access point (AP), so that direct communication is challenging. To reduce
the power required to connect to the AP, the blocked user may offer to send a
payment of power (via WPT) to a more advantageously placed UE, in exchange for
relay services. The core problem we address in this paper is the
identification of the optimal relay candidate from a potentially extensive set
of options, along with the determination of the associated power cost.

Extensive prior work considers the use of wireless power transfer and related
technologies to enable relaying in cooperative settings, in contexts such as
IoT networks or device-to-device communication [4, 5]. Other works, such as
[6], more explicitly acknowledge the need to incentivize the relay operation,
but still assume a basic level of cooperation.

Figure 1: Fundamental scenario considered in this work: A single source UE
attempts to communicate with a blocked AP using one of the $n$ relay candidate
UEs, incentivizing cooperation with wireless power transfer.

We instead consider fully non-cooperative relay candidate devices that are
energy limited, and possibly mobile (e.g., a cellular phone or autonomous
vehicle), thus requiring an incentive to provide the relay service. We
approach the scenario from a game-theoretic perspective, inspired by several-
decades-old seminal work on auction theory, to develop a protocol which
creates emergent cooperation among non-cooperative entities.

Game theory has long been an important tool in the design and analysis of
communication systems [7], and recent years have seen the use of auctions
linked to WPT in conjunction with energy markets [8]. These works, however,
apply auctions in the fiscal domain, using money as the payment medium. In
contrast, we propose the use of wireless power itself as an RF-native
“currency”, thus eliminating the need for complex configuration and
integration between the physical wireless system and financial infrastructure.

Our contributions are succinctly stated below:

  * •

We consider a source UE with a non-line-of-sight (NLoS) channel to an AP, so
that communication either requires significant transmit power or is impossible
altogether. This source seeks to use WPT to incentivize non-cooperative,
energy limited UEs to act as cooperative relays (see Fig. 1).

  * •

We pose the problem of finding a cooperation-inducing protocol which minimizes
the energy consumption of the source, while accounting for the self-interested
rationality of the relay candidates and their private channel state
information, which is unavailable to the source.

  * •

We show that the solution to our problem is given by a Myerson auction [9],
with valuations determined by the minimum source transmit power needed for the
relay candidates to be neutral towards cooperation. We provide a rigorous
proof that these valuations are regular for wireless channels described by
lognormal fading, which allows the optimal source transmit power to be found
with a simple bisection search.

  * •

We validate our theoretical findings through extensive simulation experiments,
with communication and WPT parameter values taken from real-world experiments
reported in the literature. Our results demonstrate a 91% reduction in outage
probability with as few as 4 relay candidates compared to the non-cooperative
alternative, and as much as 48% source power savings compared to a baseline
approach.

##  II System Model and Motivation

We consider a system consisting of a single source UE, $n$ battery-powered
relay candidate UEs, and a single AP, as depicted in Fig. 1. We assume that
the channel between the source and the AP is quite poor, e.g., due to a LoS
blockage, so that direct communication with the AP is either impossible or
requires an unreasonably large transmit power. Consequently, the source looks
to induce one of the candidate UEs to act as its relay. Without loss of
generality, we set the AP as the origin, and let $q_{s}$ and $q_{i},\forall
i\in N=\\{1,...,n\\}$ denote the 2D positions of the source and the
$i^{\text{th}}$ candidate, respectively. We next describe our modeling of
communication and WPT.

###  II-A Communication System Model

To satisfy a given quality-of-service (QoS) requirement, e.g., maximum bit
error rate at a minimum spectral efficiency, all communication links must
maintain a minimum signal-to-noise ratio (SNR) of $\gamma_{\text{th}}$.

The SNR across any link is given by $\gamma=PH/\sigma^{2}$, where $P$ is the
transmit power, $H$ is the channel coefficient (given by the square of the
modulus of the complex channel gain between the transmitter and receiver), and
$\sigma^{2}$ is the noise power. We denote the channel coefficient between the
source (candidate) UE and the AP as $H_{s}$ ($H_{i}$), while $H_{si}$ denotes
the channel between the source and candidate $i$.

Based on [2, 10], we use a reference path loss channel model with lognormal
small scale fading, and we account for large scale fading by considering
different sets of channel parameters for LoS and NLoS cases. In the dB scale,
this model is parameterized by a path loss intercept,
$H_{\text{PL},0,\text{dB}}^{c}$, a path loss exponent, $n_{\text{PL}}^{c}$,
and fading power, $(\sigma_{\text{SS, dB}}^{c})^{2}$, where
$c\in\\{\text{L},\,\text{NL}\\}$ indicates the corresponding channel
parameters for the LoS and NLoS cases. We assume that $H_{i}$ and $H_{si}$ are
LoS channels, as depicted in Fig. 1, as otherwise, the candidate would almost
certainly not be useful as a relay. We thus express these channels as

$H_{l,\text{dB}}=H_{\text{PL},0,\text{dB}}^{\text{L}}-10n_{\text{PL}}^{\text{L}}\text{log}_{10}(d_{l})-H_{\sigma_{\text{SS,dB}}^{\text{L}},l},\vspace{-0.025in}$

for $l\in\cup_{i=1}^{n}\\{si,\,i\\}$, with $d_{l}$ the corresponding distance
between the two endpoints of the channel and
$H_{\sigma_{\text{SS,dB}}^{\text{L}},l}\sim\mathcal{N}(0,\,(\sigma_{\text{SS,dB}}^{\text{L}})^{2})$.
We also assume that $H_{s}$ is NLoS:

$H_{s,\text{dB}}=H_{\text{PL},0,\text{dB}}^{\text{NL}}-10n_{\text{PL}}^{\text{NL}}\text{log}_{10}(d_{s})-H_{\sigma_{\text{SS,dB}}^{\text{NL}},s},\vspace{-0.025in}$

where $d_{s}=||q_{s}||_{2}^{2}$ is the distance between the source and the AP,
and
$H_{\sigma_{\text{SS,dB}}^{\text{NL}},s}\sim\mathcal{N}(0,\,(\sigma_{\text{SS,dB}}^{\text{NL}})^{2})$.
The small scale fading is taken to be independent across all channels [10].

If the source attempts to directly communicate with the AP, it must use a
power of $P_{s}=\gamma_{\text{th}}\sigma^{2}/H_{s}$. However, due to the NLoS
channel, this power may be quite large and may exceed the maximum transmit
power of the device, $P_{\text{max}}$, causing an outage. Alternatively, if
the transmission is cooperatively relayed via candidate $i$, the source first
transmits by expending $P_{si}=\gamma_{\text{th}}\sigma^{2}/H_{si}$, after
which the candidate transmits with power
$P_{i}=\gamma_{\text{th}}\sigma^{2}/H_{i}$ over the LoS link with the AP.

Under certain channel conditions (specifically, if
$H_{si}^{-1}+H_{i}^{-1}<H_{s}^{-1}$) this relaying setup provides improved
energy-efficient connectivity. However, the gains from cooperation are not
equally distributed across the source and the chosen relay candidate, as the
candidate expends energy from its limited store to help meet the QoS
requirement of the source. Consequently, in scenarios where the source and
relay are independent, non-cooperative actors, the candidate has no incentive
to participate in this scheme.

###  II-B Wireless Power Transfer

In order to achieve the utility transfer necessary to enable candidate
participation in the aforementioned scenario, we propose leveraging wireless
power transfer from the source to the relay candidates. More specifically, the
source uses simultaneous wireless information and power transfer (SWIPT), so
that the total source transmit power is given by
$P^{\text{tot}}=P_{si}+P_{\text{WPT}}$, where $P_{\text{WPT}}$ is the power
allocated for WPT. The candidate then uses a dynamic power-splitting setup
[11], with the power splitting ratio given by $P_{\text{WPT}}/P^{\text{tot}}$.
The power harvested by a candidate relay $i$ is then given by
$P_{i}^{\text{harv}}=\alpha P_{\text{WPT}}A_{r}H_{si}$ [12], with
$\alpha\in[0,1]$ denoting the efficiency of the power harvesting circuitry,
and $A_{r}$ denoting the effective aperture area of the receiving antenna
(assumed to be identical for all candidates).

The candidate will only participate in the relay scheme if it harvests at
least $P_{i}$ power, which covers the subsequent transmission to the AP. Thus,
the minimum source transmit power that induces cooperation with candidate $i$,
is

$P_{si}^{\text{min}}=P_{si}+\frac{P_{i}}{\alpha A_{r}H_{si}}.$ (1)

If $P^{\text{tot}}\geq P_{si}^{\text{min}}$, the candidate harvests
$P_{i}^{\text{harv}}=\alpha(P^{\text{tot}}-P_{si})A_{r}H_{si}\geq P_{i}$, and
any surplus power can be used for charging its battery or powering on-board
sensors. However, $P_{si}^{\text{min}}$ is not exactly known to the source, as
we next detail in our information availability assumptions.

###  II-C Information Assumptions

We assume that both the source and the relay candidates individually know the
value of the channel coefficient between themselves and the AP, but do not
have precise information about the channel between other UEs and the AP.
Additionally, the channel coefficient between the source and a candidate is
known to both parties, but unknown to other candidates.

We also assume that the source and each candidate can estimate the channel,
$H_{i}$, between any candidate and the AP using estimation frameworks such as
[13], in conjunction with information acquired from prior data collected
during operation, including the coordination process presented in Section III.
Specifically, the channel parameters $H_{\text{PL},0,\text{dB}}^{\text{L}}$,
$n_{\text{PL}}^{\text{L}}$ and $\sigma_{\text{SS,dB}}^{\text{L}}$, along with
candidate locations, $q_{i}$, and source location $q_{s}$, are known to all,
but the specific realization of the fading component
$H_{\sigma_{\text{SS,dB}}^{\text{L}},i}$ is known only to candidate $i$.
Analyzing the impact of errors and differences in parameter estimation is an
important avenue for future exploration.

##  III Problem Formulation

We now mathematically state our WPT-incentivized relay scenario and construct
an optimization problem for efficient selection of a relay candidate. Our
objective is to design a protocol, known beforehand to all UEs, that
incentivizes cooperation between the source and candidate UEs, while adhering
to informational limitations. Fundamentally, the protocol should ensure that
the source is able to communicate with the AP whenever possible. Additionally,
as the source initiates the process, our protocol prioritizes minimizing its
overall energy consumption.

To achieve this, we propose the following streamlined coordination procedure:
First, the source broadcasts a signal111This signal can also be used to
estimate channels, $H_{si}$, and locations $q_{i}$, by employing integrated
sensing and communication (ISAC) or other techniques. that queries for
feasible relay candidates for data transmission. Each candidate then responds
by indicating the amount of power it requires to serve as a relay for the
source, which, in general, may differ from the true value,
$P_{si}^{\text{min}}$. The source then decides on which candidate to use as a
relay, if any, and the total transmit power to use, $P^{\text{tot}}$. We next
model this coordination procedure as an auction game and formally state our
optimization objective to make the WPT-driven relay scenario power-efficient
for the source.

###  III-A Bayesian Reverse Auction Formulation

A reverse auction [14] is a procurement process in which bidders compete to
win a contract specified by a buyer. Bidders submit bids,
$b=\\{b_{i}\\}_{i=1}^{n}$, to the buyer, who then decides which bidder,
determined by a function $\rho(b)$, wins the auction, and the amount to pay,
denoted by $x(b)$. The pair $(\rho,\,x)$ is referred to as the auction
mechanism [9], and is itself a function of the buyer’s valuation, $v_{0}$,
i.e., the cost for the buyer to directly fulfill the contract (we set
$\rho(b)=0$ in such a scenario). Additionally, each bidder has a valuation,
$v_{i}$, which is conceptually its “breakeven” price point; if $x(b)<v_{i}$,
the bidder’s “cost” to fulfill the contract is less than the payment. While
the buyer’s valuation is known to all, the bidders’ valuations are private,
i.e., unknown to the buyer or each other, and thus, the bids may differ from
valuations. In a Bayesian auction, however, the valuations are drawn from
known distributions, characterized by PDFs $f_{i}(v_{i})$ with support over
$V_{i}$.

Our problem is well modeled as a Bayesian reverse auction, with the candidates
as bidders and the source as the buyer. The private valuation of each
candidate is given by $v_{i}=P_{si}^{\text{min}}$, the minimum source transmit
power which could induce cooperation, while the valuation of the source is
$v_{0}=\min{}\\{P_{\text{max}},\,P_{s}\\}$, which is communicated to all
candidates in the coordination protocol222This is necessary to avoid market
manipulation by the source. Further study of trust mechanisms in our
communication system is an important direction for future work.. If $P_{s}\leq
P_{\text{max}}$, then $v_{0}$ represents the power required for the source to
communicate directly with the AP. Otherwise, $v_{0}$ is a placeholder bid
indicating the maximum amount the source can pay. Furthermore,
$x(b)=P^{\text{tot}}\in[0,P_{\max}]$ determines the price the source pays
expressed as the total transmit power for both communication and WPT (if
$\rho(b)\neq 0$).

The source’s knowledge of the channel statistics, $H_{i},\,\forall i\in N$,
provides a distribution over valuations $P_{si}^{\text{min}}$. More
specifically, $P_{si}^{\text{min}}$ is a function of the realization of the
channel $H_{i}$, known to candidate $i$ at the time of placing the bid.
Decomposing $H_{i}$ into the deterministic path loss component,
$H_{\text{PL},i}$ and a stochastic small-scale fading component,
$H_{\text{SS},i}\sim\text{Lognormal}(0,\sigma_{\text{SS}}^{\text{L}})$, we
have

$P_{si}^{\text{min}}=P_{si}+\frac{\gamma_{\text{th}}\sigma^{2}}{H_{\text{PL},i}H_{\text{SS},i}\alpha
A_{r}H_{si}}.$

Therefore, the PDF of $P_{si}^{\text{min}}$, denoted by
$f_{i}^{\text{min}}(\cdot)$, is given as

$\begin{split}f_{i}^{\text{min}}(P_{si}^{\text{min}})&=f_{\text{ln},\sigma_{SS}^{\text{L}}}\big{(}H_{\text{SS},i}(P_{si}^{\text{min}})\big{)}\left|\frac{dH_{\text{SS},i}(P_{si}^{\text{min}})}{dP_{si}^{\text{min}}}\right|\\\
&=f_{\text{ln},\sigma_{SS}^{\text{L}}}\big{(}H_{\text{SS},i}(P_{si}^{\text{min}})\big{)}\frac{H_{\text{SS},i}(P_{si}^{\text{min}})}{(P_{si}^{\text{min}}-P_{si})},\\\
\text{with
}H_{\text{SS},i}&(P_{si}^{\text{min}})=\frac{\gamma_{\text{th}}\sigma^{2}}{(P_{si}^{\text{min}}-P_{si})H_{\text{PL},i}\alpha
A_{r}H_{si}},\end{split}$ (2)

where $f_{\text{ln},\sigma_{SS}^{\text{L}}}(\cdot)$ is the PDF of
$H_{\text{SS},i}$. We denote the support of $P_{si}^{\text{min}}$ as
$\mathcal{P}_{i}$, and the support over the joint valuation space is given by
$\mathcal{P}=\prod_{i=0}^{n}\mathcal{P}_{i}$.

We do not disallow bids that differ from the valuations, but the revelation
principle [14] states that for any auction with untruthful bids, there is an
analogous auction with truthful bids that produces the same outcome.
Therefore, without loss of generality, we may focus on the case where the
candidates bid their true valuations, as long as we also ensure that honest
bidding of valuations is optimal for all candidates, as we do in the sequel.
Thus, a candidate’s bid is given by its valuation, i.e.,
$b_{i}=v_{i}=P_{si}^{\text{min}}$.

###  III-B Formal Problem Statement

The problem of minimizing the transmit power of the source through a choice of
the mechanism $(\rho,x)$ can be formally expressed as follows: —s—[2] ρ, x
E_v∼P [x(v)] E_v_-i∼P_-i[1_i=ρ(v)( x(v) - v_i)] ≥0, ∀i∈N, v_i∈P_i,
E_v_-i∼P_-i[1_i=ρ(b)( x(v) - v_i)] ≥ E_v_-i∼P_-i[1_i=ρ(v_-i, s_i)( x(v_-i,
s_i) - v_i)], ∀i∈N, ∀v_i∈P_i, ∀s_i∈P_i. where $v=\\{v_{i}\\}_{i=0}^{n}$,
$v_{-i}$ denotes the set of all valuations except $v_{i}$, $(v_{-i},s_{i})$
indicates a joint valuation profile $v$ with $v_{i}$ replaced by $s_{i}$,
$\mathds{1}_{\mathcal{C}}$ is a binary indicator of whether condition
$\mathcal{C}$ is met, and $\mathbb{E}_{v\sim\mathcal{P}}[x(v)]$ is referred to
as the expected revenue. Constraint (III-B) enforces that no candidate expects
to expend more energy by participating in the auction than they would
otherwise (individual rationality), and Constraint (III-B) ensures that
bidding the true valuation is an optimal strategy for all candidates
(incentive compatibility), as required per our application of the revelation
principle. In the auction context, this problem corresponds to designing a
revenue-maximizing auction, as described in the next section.

The interpretation of the auction result in the context of the underlying
communication system requires brief consideration. In particular, when
$\rho(v)=0$, the source “wins” the auction (no suitable candidate is found),
and if, additionally, $P_{s}\leq P_{\text{max}}$, the source communicates
directly with the AP with power $x(v)=P_{s}$. However, if
$P_{s}>P_{\text{max}}$, the source will be unable to connect to the AP, and
$x(v)=P_{\text{max}}$ is no longer representative of the transmit power (which
is $0$), as no communication occurs. On the other hand, any mechanism that
does not result in communication when feasible can be shown to be sub-optimal
within the auction framework333A simple proof proceeds by contradiction but is
omitted for brevity.. Further, when communication is infeasible, i.e., if
$P_{\text{max}}<\min\bigl{(}\\{P_{s}\\}\cup\\{P_{si}^{\text{min}}\\}_{i\in
N}\bigr{)}$, all mechanisms that respect individual rationality yield the same
result, $x(v)=P_{\text{max}}$, so that the optimization is effectively over
only scenarios where connectivity is achievable. Thus, optimality within the
auction framework corresponds to a protocol in the communication system which
(1) always results in connectivity when feasible and (2) minimizes the
expectation of source’s transmit power conditioned on the event that
connectivity is possible.

Perfect Information Lower Bound: To establish a benchmark for evaluation, we
introduce a baseline lower bound for comparison. The perfect information
solution minimizes the source’s power while respecting incentive
compatibility, but ignores informational constraints, i.e., the source has
perfect knowledge of $H_{i}$ for all $i$. This is denoted as
$P^{\text{lb}}=\min_{i\in N}P_{si}^{\text{min}}$, and corresponds to the
source making a take-it-or-leave-it offer of $P^{\text{lb}}-\epsilon$ for an
arbitrarily small $\epsilon$, which will be accepted by
$i^{\text{lb}}=\text{arg}\min_{i\in N}P_{si}^{\text{min}}$.

##  IV Solution via Myerson Auctions

To solve the optimization problem of (III-B), we employ the seminal results
presented in a several-decade-old work on auction theory to design the
mechanism that maximizes the expected revenue of the source, while respecting
individual-rationality and incentive-compatibility constraints. In [9],
Myerson demonstrated the construction of such an optimal mechanism for forward
auctions when the valuation distributions of the bidders satisfy certain
technical constraints. We next show how to extend the optimal Myerson
mechanism to reverse auctions, and then prove that our scenario satisfies the
corresponding technical constraints.

###  IV-A Buyer-Optimal Reverse Auctions

To describe the optimal auction, we first introduce the virtual valuation,
$c_{i}(v_{i})$, which for a reverse auction we define as follows:

$c_{i}(v_{i})=v_{i}+\frac{F_{i}(v_{i})}{f_{i}(v_{i})}$

where $F_{i}(v_{i})$ and $f_{i}(v_{i})$ are the CDF and PDF of $v_{i}$,
respectively. We call $f$ a regular distribution if
$dc_{i}(v_{i})/dv_{i}>0,\,\forall v_{i}\in V_{i}$, i.e., the virtual valuation
is strictly increasing in the valuation over the support. We now state the key
result:

######  Lemma 1.

If all bidder valuations are independently distributed according to regular
distributions, $f_{i}$, then for an auction with valuations
$v=\\{v_{i}\\}_{i=0}^{n}$, the revenue-maximizing mechanism for a reverse
auction is given by

$\rho^{*}(v)=\begin{cases}0&\text{If }v_{0}\ <\ c_{i}(v_{i})\ \forall i\in
N\\\ \operatorname*{arg\,min}_{i\in
N}c_{i}(v_{i})&\text{Otherwise}\end{cases}$ (3)

and

$x^{*}(v)=\begin{cases}0&\text{If }\rho^{*}(v)=0\\\
z(\rho^{*}(v))&\text{Otherwise}\end{cases}$ (4)

with $z(i)=\sup\\{s\,|\,c_{i}(s)<v_{0}\,\text{ and }\,c_{i}(s)<c_{j}(v_{j}),\
\forall j\neq i\\}$

###### Proof.

We note that the reverse auction may be transformed into a forward auction
with valuations $\\{-v_{i}\\}_{i=0}^{n}$ and payments $-x(v)$. The result then
follows from [9, Lemma 3]. ∎

Therefore, the optimal Myerson reverse auction assigns the object to the
bidder, $\rho(v)=i^{*}$, with the lowest virtual valuation,
$c_{i^{*}}(v_{i^{*}})$, or the buyer, if its valuation, $v_{0}$, is lower than
all virtual valuations. The payment $x^{*}(v)$ is the highest valuation that
would still result in $i^{*}$ winning the auction, which motivates the
placement of honest bids from the participants. We note that while, in
general, a closed form expression for $z$ cannot be found, a simple bisection
search can quickly find a solution, given the monotonicity of $c_{i}$.

###  IV-B Optimal WPT Auction

Figure 2: Sample simulation results: (a) Consistent reduction in outage
probability with a Myerson auction-based relay selection, incentivized by WPT
for (left) various values of WPT circuitry efficiency, $\alpha$ with
$\Gamma=1$ and (right) several scales of small-sacle fading variance, $\Gamma$
with $\alpha=0.2$. (b) Higher relative source power savings of the Myerson
mechanism with increasing number of relay candidates (c) Variation in
harvested power with WPT efficiency ($\alpha$) as the number of relay
candidates grow, with an overall reduction due to higher source power savings.

To apply Lemma 1, we must prove that the valuations of the candidates,
$v_{i}=P_{si}^{\text{min}}$, are regular. Using the dependence on the log-
normally distributed fading component of the channel developed in Eq. (2), we
show in the following lemma that this is indeed the case.

######  Lemma 2.

The distribution for valuations $v_{i}=P_{si}^{\text{min}}$, as given by Eq.
(2), is regular.

###### Proof.

From Eq. (2), it can be shown that

$\begin{split}\frac{dc_{i}(v_{i})}{dv_{i}}&=2-\frac{F_{i}(v_{i})}{f_{i}(v_{i})^{2}}\frac{df_{i}(v_{i})}{dv_{i}}\\\
&=2+\frac{1-F_{\text{ln},\sigma_{SS}^{\text{L}}}(H_{\text{SS},i}(v_{i}))}{f_{\text{ln},\sigma_{SS}^{\text{L}}}(H_{\text{SS},i}(v_{i}))H_{\text{SS},i}(v_{i})}\\\
&\quad\quad\quad\quad\times[1-\frac{\text{ln}(H_{\text{SS},i}(v_{i}))}{(\sigma_{\text{SS}}^{\text{L}})^{2}}]\\\
&=2+\frac{1-\Phi(z)}{\phi(z)}[\sigma_{SS}^{\text{L}}-z]\end{split}$

where $\Phi$ and $\phi$ are, respectively, the CDF and PDF of the standard
normal distribution, and
$z=\text{ln}(H_{\text{SS},i}(v_{i}))/(\sigma_{\text{SS}}^{\text{L}})$. By
inspection, the above expression is positive for $z\leq 0$. For $z>0$, we note
that the ratio $r(z)=\big{(}1-\Phi(z)\big{)}/\phi(z)$ is the Mills ratio for
the standard normal distribution, and it can be shown that $r(z)<1/z,\;\forall
z>0$, [15]. Therefore,

$2+\frac{1-\Phi(z)}{\phi(z)}[\sigma_{\text{SS}}^{\text{L}}-z]>2-\frac{1-\Phi(z)}{\phi(z)}z>1,\,\forall
z>0.\vspace{-0.1in}$

∎

Having shown the regularity of the bidders’ valuations, we can now apply the
optimal mechanism to our reverse auction:

######  Corollary 1.

The optimal solution to (III-B) is given by Eqs. (3) and (4) with
$v_{0}=P_{s},\,v_{i}=P_{si}^{\text{min}},\,\forall i\in N$, and
$f_{i}(v_{i})=f_{i}^{\text{min}}(P_{si}^{\text{min}})$, as given by Eq. (2).

###### Proof.

This follows from Lemmas 1 and 2. ∎

######  Remark 1.

While the above results have been derived for lognormal fading, similar
results may be derived for other statistical channel models, even if they are
not regular. However, if not regular, a more complex method must be used [9],
which may be less computationally tractable.

##  V Numerical Examples

To verify the validity of our proposed approach, we run a number of simulation
experiments with system parameters as reported from experimental studies. In
particular, we consider the scenario depicted in Fig. 1, in which a source
seeks to connect with an access point with which it has a NLoS channel. We
randomly place up to 10 relay candidates with LoS channels to both the AP and
the source. We then follow the auction protocol described in Section IV to
select a candidate relay (if any) and determine the transmit power of the
source. Throughout, we assume all links communicate using 8-QAM modulation,
with a target bit error rate (BER) of $1\mathrm{e}{-6}$, so that the target
SNR in dB is given by $\gamma_{\text{th,dB}}=33.18$ [16]. Communication noise
is modeled as AWGN with power $\sigma^{2}=-75\,$dBm, and the maximum transmit
power of the source is $100\,$mW. The channel parameters are given as follows:
$H_{\text{PL},0,dB}^{\text{L}}=0$, $H_{\text{PL},0,dB}^{\text{NL}}=-25$,
$n_{\text{PL}}^{\text{L}}=2.5$, $n_{\text{PL}}^{\text{NL}}=5.76$,
$\sigma_{SS,dB}^{\text{L}}=8.66$, and $\sigma_{SS,dB}^{\text{NL}}=9.06$ [2,
Table VI], [17]. For WPT, we take $\alpha=0.2$ and $A_{r}=1\,\text{cm}^{2}$.
For several experiments, we vary both $\sigma_{SS,dB}^{\text{L}}$ and
$\sigma_{SS,dB}^{\text{NL}}$ by scaling the given values with a factor,
$\Gamma\in\\{0.2,0.6,1.0,1.4\\}$. Each reported average is taken over a set of
10,000 simulations with randomly sampled channel fading, and, unless stated
otherwise, random placement of the candidates.

We next examine system performance in terms of outage probability, source
transmit power, and harvested power, while exploring system parameters
including circuitry efficiency, $\alpha$, fading power,
$\sigma_{SS,dB}^{\text{L}}$, and number of candidates, $n$.

###  V-A Outage Probability

We first study the impact of our protocol on the outage probability of the
source. In the absence of relay candidates, the source experiences an outage
probability of $96\,$%. Fig. 2 (a) plots the outage probability of the source
for different values of $n$, $\alpha$ and fading scale factors, $\Gamma$. We
see that the presence of even a single relay candidate significantly reduces
the outage probability to as little as $45\,$%, and this probability drops to
near $0\,$% for higher values of $n$. Furthermore, increasing either power
efficiency, $\alpha$, or fading factor, $\Gamma$, reduces outage probability.

###  V-B Source Transmit Power

We next examine the transmit power used by the source as determined by the
auction protocol. To gauge energy efficiency, we compare the transmit power,
$P^{tot}$, of the source under the Myerson mechanism to the perfect-
information transmit power, $P^{\text{lb}}$ (described in Section III-B),
which is a lower bound on $P^{\text{tot}}$. As an upper bound, we also compare
against a standard sealed-bid, second-price auction (Vickrey auction), in
which the source selects the candidate relay with the lowest valuation but
transmits with a power equal to the second-lowest valuation [18]. Fig. 2 (b)
shows significant power savings when using the proposed buyer-optimal Myerson
auction protocol compared to a standard Vickrey auction, particularly when
fewer relay candidates are present. Furthermore, as the number of candidates
increase, our proposed Myerson auction solution approaches the perfect-
information lower bound. We also note that the performance of the Vickrey and
Myerson auctions become comparable when the number of candidates is large,
suggesting that in scenarios where candidates and sources have less
information about each other’s channels, the Vickrey auction may produce
acceptable results given a sufficiently high density of UEs.

Figure 3: (a) Sample placement of UEs for candidate selection study (b)
Variation in selection likelihood across several scales ($\Gamma$) of fading.
See color PDF for optimal viewing.

###  V-C Harvested Transmission Power

While energy harvesting is not the chief objective of our protocol, it plays
an important role in motivating the candidates to cooperate. Fig. 2 (c) shows
the average amount of power harvested by candidates for various efficiency
factors, $\alpha$. As expected, more energy is harvested as the WPT efficiency
is increased, and we see that the candidates receive small but significant
boosts on the order of tens of $\mu$W. However, as the number of candidates
grow, competition allows the source to reduce the transmit power,
$P^{\text{tot}}$, which in turn reduces the power available for harvesting.

###  V-D Candidate Assignment

To better understand how the placement of a candidate affects its likelihood
of winning the auction, we consider the system of $n=4$ candidates at fixed
positions, as shown in Fig. 3 (a). For each value of $\Gamma$, we sample
10,000 realizations of channel fading and report the frequency with which each
candidate is chosen, as shown in Fig. 3 (b). At low fading, we see that
Candidate 1, whose proximity to the source results in more efficient WPT,
consistently wins the auction. However, as the fading power increases, the
location-dependent path loss component of the channel loses prominence, and
the auction mechanism picks the other candidates more frequently.

##  VI Conclusion

This paper addresses the use of wireless power transfer (WPT) to incentivize
energy-constrained user equipment (UE) to serve as cooperative relays in
challenging non-line-of-sight communication scenarios. By proposing a
cooperation-inducing protocol inspired by Myerson auction theory, we minimize
the expected power used by the source while considering the self-interest of
relay candidates and their private channel state information. Rigorous
analysis establishes the regularity of valuations for channels modeled with
lognormal fading, facilitating efficient determination of the optimal source
transmit power. Extensive simulation experiments validate our framework,
demonstrating a significant reduction in outage probability and power
consumption, thus promoting energy-efficient cooperative relaying in wireless
environments.

## References

  * [1] ITU-R, “Framework and overall objectives of the future development of IMT for 2030 and beyond,” ITU-R, Tech. Rep., 2023, Draft New Recommendation.
  * [2] I. A. Hemadeh, K. Satyanarayana, M. El-Hajjar, and L. Hanzo, “Millimeter-Wave Communications: Physical Channel Models, Design Considerations, Antenna Constructions, and Link-Budget,” _IEEE Communications Surveys & Tutorials_, vol. 20, no. 2, pp. 870–913, 2018.
  * [3] M. A. Hossain, R. Md Noor, K.-L. A. Yau, I. Ahmedy, and S. S. Anjum, “A Survey on Simultaneous Wireless Information and Power Transfer With Cooperative Relay and Future Challenges,” _IEEE Access_ , vol. 7, pp. 19 166–19 198, 2019.
  * [4] X. Li, Q. Wang, M. Liu, J. Li, H. Peng, M. J. Piran, and L. Li, “Cooperative Wireless-Powered NOMA Relaying for B5G IoT Networks With Hardware Impairments and Channel Estimation Errors,” _IEEE Internet of Things Journal_ , vol. 8, no. 7, pp. 5453–5467, 2021.
  * [5] M. M. Salim, H. A. Elsayed, M. S. Abdalzaher, and M. M. Fouda, “RF Energy Harvesting Dependency for Power Optimized Two-Way Relaying D2D Communication,” in _2022 IEEE Int. Conf. on Internet of Things and Intelligence Systems (IoTaIS)_ , 2022, pp. 297–303.
  * [6] Y. Xu, J. Liu, H. Takakura, Z. Li, Y. Ji, and N. Shiratori, “Stackelberg Game-based Secure Communication in SWIPT-enabled Relaying Systems,” in _ICC 2022 - IEEE Int. Conf. Commun._ , 2022, pp. 1462–1467.
  * [7] Z. Han, D. Niyato, W. Saad, T. Basar, and A. Hjorungnes, _Game Theory in Wireless and Communication Networks: Theory, Models, and Applications_. Cambridge University Press, 2012.
  * [8] Y. Ni, S. Zhou, Q. Wang, Y. Zhou, and H. Zhu, “Auction Game Based Phone-to-Phone Electricity Trading via Wireless Energy Transfer,” in _2019 IEEE 19th Int. Conf. Commun. Technol (ICCT)_ , 2019, pp. 1213–1219.
  * [9] R. B. Myerson, “Optimal Auction Design,” _Mathematics of Operations Research_ , vol. 6, no. 1, pp. 58–73, 1981.
  * [10] M. K. Samimi, G. R. MacCartney, S. Sun, and T. S. Rappaport, “28 GHz Millimeter-Wave Ultrawideband Small-Scale Fading Models in Wireless Channels,” in _2016 IEEE 83rd Vehicular Technology Conference (VTC Spring)_ , 2016, pp. 1–6.
  * [11] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: Architecture design and rate-energy tradeoff,” in _2012 IEEE Global Communications Conf. (GLOBECOM)_ , 2012, pp. 3982–3987.
  * [12] L. Liu, R. Zhang, and K.-C. Chua, “Wireless Information and Power Transfer: A Dynamic Power Splitting Approach,” _IEEE Transactions on Communications_ , vol. 61, no. 9, pp. 3990–4001, 2013.
  * [13] M. Malmirchegini and Y. Mostofi, “On the Spatial Predictability of Communication Channels,” _IEEE Trans. on Wireless Commun._ , 2012.
  * [14] N. Noam, T. Roughgarden, E. Tardos, and V. V. Vazirani, _Algorithmic Game Theory_. Cambridge University Press, 2007.
  * [15] R. D. Gordon, “Values of Mills’ Ratio of Area to Bounding Ordinate and of the Normal Probability Integral for Large Values of the Argument,” _The Ann. of Mathematical Statistics_ , vol. 12, no. 3, pp. 364–366, 1941.
  * [16] A. Goldsmith, _Wireless Communications_. Cambridge Univ. Press, 2005.
  * [17] Y. Azar, G. N. Wong, K. Wang, R. Mayzus, J. K. Schulz, H. Zhao, F. Gutierrez, D. Hwang, and T. S. Rappaport, “28 GHz propagation measurements for outdoor cellular communications using steerable beam antennas in New York city,” in _2013 IEEE Int. Conf. on Commun. (ICC)_ , 2013, pp. 5143–5147.
  * [18] W. Vickrey, “Counterspeculation, Auctions, and Competitive Sealed Tenders,” _The Journal of Finance_ , vol. 16, no. 1, pp. 8–37, 1961.