Box analogue of the Bourgain–Chang multilinear Burgess bound: power saving for $\chi(\prod_{j=1}^n L_j(\mathbf{x}))$ over axis-parallel boxes of volume exceeding $p^{n/4+\varepsilon}$

Bourgain and Chang (2010) proved that if $\chi$ is a nontrivial multiplicative character modulo a large prime $p$ and $L_1,\dots,L_n$ are linearly independent linear forms over $\mathbb{F}_p$, then the multilinear sum $\sum_{\mathbf{x}}\chi(\prod_{j=1}^n L_j(\mathbf{x}))$ over a cube of side $H>p^{1/4+\varepsilon}$ admits a power saving $p^{-\delta}$ over the trivial bound, uniformly in the forms and the character. In a closing remark they asked for the box analogue: the same power saving over an arbitrary axis-parallel box of side lengths $H_1,\dots,H_n$ satisfying only the volumetric condition $\prod_i H_i>p^{n/4+\varepsilon}$, under which individual sides may be far shorter than $p^{1/4}$; their methods, they noted, settle only small $n$. Later work has advanced the surrounding theory — Pierce–Xu for generic forms at weaker per-side thresholds, Chu (2025) for forms splitting over the algebraic closure with every side at least $p^{1/4+\kappa}$, and Gabdullin and Chattopadhyay for the related Davenport–Lewis problem over $\mathbb{F}_{p^n}$ — but the known bounds for polynomial-argument character sums that reach below the Burgess threshold on individual sides require the number of variables to be large relative to the degree, so the multilinear box problem is untouched. The problem asks for a proof, or a disproof, of the volumetric box estimate for the multilinear character sum in arbitrary dimension.

Background

The study of short character sums is a central topic in analytic number theory. Fix a prime $p$ and a nontrivial multiplicative (Dirichlet) character $\chi$ modulo $p$, extended to all of $\mathbb{F}_p$ by the convention $\chi(0)=0$. For an interval of $H$ consecutive integers one wants upper bounds for $|\sum_{x=N+1}^{N+H}\chi(x)|$ that beat the trivial bound $H$. Burgess's classical inequality (1962) gives a power saving: for every $\varepsilon>0$ there is $\delta>0$ such that the sum is $\ll p^{-\delta}H$ as soon as $H>p^{1/4+\varepsilon}$, uniformly in $N$ and $\chi$. The exponent $1/4$ (the Burgess threshold) is the natural barrier of the method and has essentially not been improved in over sixty years.

A multivariate generalization replaces the linear argument of $\chi$ by a product of linear forms. Let $L_1,\dots,L_n$ be linear forms in the variables $x_1,\dots,x_n$ over $\mathbb{F}_p$ forming a nondegenerate system — that is, the $n\times n$ coefficient matrix is invertible over $\mathbb{F}_p$ — and consider

$$S = \sum_{\mathbf{x}\,\in\,\prod_{i=1}^n [a_i,\,a_i+H]} \chi\Big(\prod_{j=1}^{n} L_j(\mathbf{x})\Big),$$

where the integer vector $\mathbf{x}$ ranges over a cube of side $H$ and the arguments are reduced modulo $p$. For $n=2$ the argument of $\chi$ is a binary quadratic form: Burgess (1968) obtained a nontrivial bound for $H>p^{1/3+\varepsilon}$ and stated without proof an extension to products of $n$ linearly independent linear forms at the weaker threshold $H>p^{1/2-1/(2(n+1))+\varepsilon}$. Chang (2009) reached the Burgess threshold $p^{1/4+\varepsilon}$ for $n=2$, in both the irreducible and reducible cases. Bourgain and Chang (2010) settled the general case for cubes: if $H>p^{1/4+\varepsilon}$ then $|S|<p^{-\delta}H^n$ with $\delta=\delta_n(\varepsilon)>0$, uniformly in the system of forms, the character, and the cube's location; for $n=1$ this recovers Burgess's theorem.

The proof of Bourgain–Chang follows the Burgess amplification scheme — shifting $\mathbf{x}\mapsto\mathbf{x}+t\mathbf{y}$ and averaging over the shift — which reduces the estimate to a bound on a multiplicative energy: the number of quadruples $(\mathbf{x},\mathbf{y},\mathbf{x}',\mathbf{y}')$ in suitable cubes with $L_j\mathbf{x}\cdot L_j\mathbf{y}\equiv L_j\mathbf{x}'\cdot L_j\mathbf{y}' \pmod p$ for all $j$. (In general, the multiplicative energy of a set $A$ is the number of solutions of $ab=a'b'$ with $a,a',b,b'\in A$.) Their new ingredient is an estimate for this energy via the geometry of numbers: the solution sets are points of a lattice, controlled by Minkowski's second theorem — the fact that for a lattice $\Lambda\subset\mathbb{R}^d$ and a symmetric convex body, the product of the successive minima $\lambda_1\le\dots\le\lambda_d$ (the smallest scale factors at which the body contains $1,2,\dots$ linearly independent lattice vectors) is comparable to the lattice covolume (the volume of the lattice's fundamental domain) — together with lattice duality, the passage from a lattice to its dual lattice of vectors having integral inner product with every lattice vector. The argument exploits the symmetry of the cube at several scales.

Two neighboring problem families set the context. First, the Davenport–Lewis problem: bound $\sum_{\mathbf{x}\in B}\chi(\sum_i x_i\omega_i)$ where $\chi$ is a nontrivial multiplicative character of the extension field $\mathbb{F}_{p^n}$ (not of $\mathbb{F}_p$) and $B$ is a box in the coordinates of a basis $\{\omega_1,\dots,\omega_n\}$ of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$; when $\chi$ factors through the field norm (the map $N:\mathbb{F}_{p^n}\to\mathbb{F}_p$ sending an element to the product of its conjugates) this becomes a mod-$p$ character evaluated at a norm form, a homogeneous polynomial that is irreducible over $\mathbb{F}_p$ but splits into conjugate linear forms over $\mathbb{F}_{p^n}$. Davenport and Lewis (1963) obtained per-side thresholds $p^{1/2-1/(2(n+1))+\varepsilon}$, improved by Chang (2008), who reached the volumetric condition $\prod_i H_i>p^{(2/5+\varepsilon)n}$ for arbitrary bases, and by Konyagin (2010), who attained the Burgess threshold $p^{1/4+\varepsilon}$ in each coordinate. Second, for mod-$p$ characters at general homogeneous forms, Pierce and Xu (2020) obtained bounds for generic forms, nontrivial only when each side exceeds $p^{1/2-1/(2(n+1))+\kappa}$, and Chu (2025) — the strongest published result for the present problem's class of sums — treats forms of degree $n$ in $n$ variables that split into linearly independent linear factors over the algebraic closure $\overline{\mathbb{F}}_p$ (a class containing both the split forms of Bourgain–Chang and the norm forms), achieving a power saving whenever every side length satisfies $H_i\ge p^{1/4+\kappa}$.

All of these results control each side of the summation box individually. Bourgain and Chang's final remark asks precisely for the volumetric version in the split (multilinear) setting, and it is this box analogue that remains unsettled.

Problem Statement

Bourgain and Chang (2010) proved their multilinear Burgess bound for cubes and wrote in their final remark: 'One expects the analogue of the theorem to remain valid when summing over arbitrary boxes $B_{H_1,\dots,H_n}=\{a_i\le x_i\le a_i+H_i\ (1\le i\le n)\}$ where $0<H_i<p$ and $\prod_{i=1}^n H_i>p^{n/4+\varepsilon}$ (it is easily seen that it suffices to consider the case $H_i\le\sqrt p$). The methods described above permit to prove such statement for small values of $n$, but the general case remains to be settled.' The problem is to settle that general case.

Precisely: fix an integer $n\ge2$ and $\varepsilon>0$. Do there exist $\delta=\delta(n,\varepsilon)>0$ and $p_0=p_0(n,\varepsilon)$ such that the following holds for every prime $p>p_0$? For every nontrivial multiplicative character $\chi$ mod $p$, every system $L_1,\dots,L_n$ of linear forms in $x_1,\dots,x_n$ whose coefficient matrix is invertible over $\mathbb{F}_p$, and every axis-parallel box $B=\prod_{i=1}^n [a_i,\,a_i+H_i]$ with integers $0<H_i<p$ satisfying the volume condition

$$\prod_{i=1}^{n} H_i > p^{\,n/4+\varepsilon},$$

the sum

$$S = \sum_{\mathbf{x}\in B}\chi\Big(\prod_{j=1}^n L_j(\mathbf{x})\Big)$$

(with arguments reduced mod $p$ and $\chi(0)=0$) satisfies

$$|S| \le p^{-\delta}\prod_{i=1}^n H_i,$$

with the estimate uniform in $\chi$, in the system of forms, and in the location of the box. For $n=1$ this is exactly the classical Burgess bound; the open content is $n\ge2$ with arbitrary, not necessarily comparable, side lengths, so that individual sides may be far shorter than $p^{1/4}$ as long as the volume condition holds. As noted in the source remark, it suffices to treat the case $H_i\le\sqrt p$ for all $i$. Thus the estimate holds for all $n\ge2$ and all $\varepsilon>0$, unless for some $n\ge2$ and $\varepsilon>0$ there exist, for arbitrarily large $p$, characters, form systems, and boxes with volume above $p^{n/4+\varepsilon}$ for which no power saving over $\prod_i H_i$ holds.

References

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