Nontrivial solutions of the shifted product equation $(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)$ for algebraic shifts: existence, optimal exponent, and uniformity

Determine the true behaviour of the number $M_\nu(h,\sigma)$ of nontrivial solutions of the shifted product equation $(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)\ne0$ with $1\le x_i,y_i\le h$, where $\sigma$ is an algebraic number of degree $d$ — the only nontrivial regime being $2\le d<\nu$. Bourgain–Garaev–Konyagin–Shparlinski proved $M_\nu(h,\sigma)\le h^{\nu-d+1}\exp(O(\log h/\log\log h))$ (Theorem 26), and Heap–Sahay–Wooley independently re-proved the same exponent with an $h^{\nu-d+1+\varepsilon}$ error. Three refinements posed in the source paper remain open: (a) classify the pairs $(d,\nu)$ for which a nontrivial solution exists for every degree-$d$ algebraic $\sigma$ and all sufficiently large $h$; (b) determine the best possible exponent of $h$ — in particular whether $\nu-d+1$ is essentially optimal — including lower-bound constructions, of which none are known for any algebraic irrational shift of degree at least 2; and (c) obtain bounds whose implied constants depend explicitly on the coefficients of the minimal polynomial of $\sigma$, as the known rational-shift bounds depend on numerator and denominator. These counts are the diagonal analysis of Hurwitz zeta moments with irrational shift, so the problem gates finer moment asymptotics there.

Background

For a fixed complex number $\sigma$, integers $\nu\ge1$ and $h\ge3$, consider the Diophantine equation

$$(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)\ne0,\qquad 1\le x_i,y_i\le h.$$

A solution is called trivial (or diagonal) if $(y_1,\dots,y_\nu)$ is a permutation of $(x_1,\dots,x_\nu)$; the trivial solutions number $\nu!h^\nu+O(h^{\nu-1})$. Bourgain, Garaev, Konyagin and Shparlinski (BGKS) introduced this equation as the integer analogue of the congruence $(x_1+s)\cdots(x_\nu+s)\equiv(y_1+s)\cdots(y_\nu+s)\pmod p$ with variables in the short interval $[1,h]$, which they studied in earlier work motivated by computer-science questions such as reconstructing a hidden shifted power (an unknown monomial in a shifted variable) from noisy data modulo a prime.

The behaviour of the equation is governed by the arithmetic nature of $\sigma$. For $\sigma=0$ it is the classical multiplication-table equation $x_1\cdots x_\nu=y_1\cdots y_\nu$, which has abundant nontrivial solutions (for $\nu=2$ the total count is asymptotic to $(12/\pi^2)h^2\log h$, a classical result quoted as Lemma 24 in the BGKS paper). At the other extreme, put $G(X)=\prod_i(X+x_i)-\prod_j(X+y_j)\in\mathbb{Z}[X]$: a solution is exactly the condition $G(\sigma)=0$, that is, $G$ is divisible by the minimal polynomial of $\sigma$ (the irreducible polynomial $P_\sigma\in\mathbb{Z}[X]$, unique up to sign, of degree $d$ having $\sigma$ as a root; $d$ is called the degree of $\sigma$). Since the leading terms of $G$ cancel, $\deg G\le\nu-1$, so if $\sigma$ is transcendental, or algebraic of degree $d\ge\nu$, then $G\equiv0$ and every solution is trivial. Nontrivial solutions can occur only for algebraic $\sigma$ of degree $d<\nu$, and BGKS proved (Lemma 25 and Theorem 26, with $M_\nu(h,\sigma)$ denoting the number of nontrivial solutions)

$$M_\nu(h,\sigma)\le h^{\nu-d+1}\exp\bigl(O_\nu(\log h/\log\log h)\bigr),\qquad 2\le d<\nu,$$

leading to the asymptotic formula $K_\nu(h,\sigma)=\nu!h^\nu+O(h^{\nu-1}\exp(O(\log h/\log\log h)))$ for every irrational $\sigma$ (Corollary 27). The factor $\exp(O(\log h/\log\log h))$ is the standard maximal order of the divisor function $\tau(n)$ (the number of positive divisors of $n$); it enters because the proof counts possible values of an auxiliary integer of size about $h^{\nu-d}$ through its divisors. The exponent $\nu-d+1$ arises by fixing one variable $x_1=v$ and bounding the number of possible values of the quotient polynomial $G/P_\sigma$ evaluated at $-v$; the implied constants depend only on $\nu$, so the bound is uniform in the shift. Heap, Sahay and Wooley later gave an independent proof of the same exponent, and Wang and Xu proved a complementary paucity result (a paucity phenomenon means almost all solutions are trivial) for products of values of integer polynomials with at least two distinct complex roots.

Beyond its intrinsic Diophantine interest, this count is the diagonal analysis of the moment problem for the Hurwitz zeta function $\zeta(s,\alpha)=\sum_{n\ge1}(n+\alpha)^{-s}$: the expected main term of $\int_T^{2T}|\zeta(1/2+it,\alpha)|^{2k}\,dt$ is governed by the trivial solutions of the equation with $\nu=k$, while the nontrivial solutions control the error term and its dependence on the shift $\alpha$. The same counting problems underpin recent moment estimates for random multiplicative functions (completely multiplicative random functions with independent random phases at the primes) evaluated along polynomial sequences.

Problem Statement

Determine the true behaviour of the number $M_\nu(h,\sigma)$ of nontrivial solutions of

$$(x_1+\sigma)\cdots(x_\nu+\sigma)=(y_1+\sigma)\cdots(y_\nu+\sigma)\ne0,\qquad 1\le x_i,y_i\le h,$$

where $\sigma$ is an algebraic number of degree $d$ with minimal polynomial $P_\sigma$, in the only nontrivial regime $2\le d<\nu$ (for $d\ge\nu$ every solution is trivial, and for rational shifts there is no paucity: the total count exceeds the trivial one by a power of $\log h$). Specifically:

(a) Existence. For which pairs $(d,\nu)$ with $2\le d<\nu$ does $M_\nu(h,\sigma)\ge1$ hold for every algebraic $\sigma$ of degree $d$ and all sufficiently large $h$ (the threshold for $h$ may depend on $\sigma$)? In particular, is $M_\nu(h,\sigma)\ge1$ for every such $\sigma$ whenever $d<\nu$?

(b) Optimal exponent. Determine the best possible value of the exponent of $h$ in the upper bound of Theorem 26: find the smallest $e(d,\nu)$ such that $M_\nu(h,\sigma)\ll_{\nu,\sigma} h^{e(d,\nu)+o(1)}$ for every algebraic $\sigma$ of degree $d$, and decide whether the BGKS exponent $\nu-d+1$ is essentially optimal — both in this pointwise sense and in the uniform sense of Theorem 26, whose implied constants depend only on $\nu$. Essential optimality is a statement about matching lower bounds — nontrivial solutions in the claimed quantity for at least one algebraic $\sigma$ of each degree $d$, or for every such $\sigma$ if that is the truth; no lower bound is currently known for any algebraic irrational $\sigma$ of degree at least 2.

(c) Dependence on the shift. Prove more precise forms of the upper bounds of Theorem 26 and Lemma 25 in which the implied constants depend explicitly on the coefficients (height) of the minimal polynomial $P_\sigma$, in the way that the known rational-shift bounds of BGKS (Lemma 22 and Lemma 23, where the bounds depend on the numerator and denominator of $\sigma=t/r$) do. Since Theorem 26 and Lemma 25 are already uniform in $\sigma$ — their implied constants depend only on $\nu$ and $d$ and are independent of the coefficients of $P_\sigma$ — merely restating those bounds does not resolve this item: what is required is an estimate in which the height of $P_\sigma$ enters explicitly, as the numerator and denominator of $\sigma=t/r$ enter in Lemma 22 and Lemma 23, rather than being absorbed into a constant.

These three parts share one objective: to replace the single coarse, shift-uniform estimate of Theorem 26 by a complete quantitative theory of the nontrivial solutions of the shifted product equation — which shifts admit them, in what number, and with what dependence on the arithmetic of the shift. The source authors describe part (c) as auxiliary to their main goal but of independent interest: all three items (a), (b) and (c) belong to the question, and describing (c) as auxiliary records the source authors' relative priority, not a narrower target.

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