A power saving for intersective polynomial differences with an exponent depending only on the degree

Introduction

For a polynomial $h \in\mathbb{Z}[x]$, consider how large a set of integers can be if no difference between two of its elements is a nonzero value of $h$. Write $\mathbb{N}_{+} = \{1,2,\ldots\}$, $[N] = \{1,\ldots,N\}$, and $A-A = \{a-b : a,b \in A\}$. We call $h$ intersective if, for every integer $q \ge1$, some integer $t$ satisfies $h(t) \equiv0 \pmod q$. This is the necessary local condition for polynomial differences in every set of positive density: if $h$ has no root modulo $q$, the multiples of $q$ contain no difference in $h(\mathbb{N}_{+})$. The condition allows the roots at different moduli to arise from different factors of $h$; a rational root is not required.

Theorem 1.1. For every integer $k \ge2$, there is a constant $c_k \in(0,1)$ with the following property. For every intersective polynomial $h \in\mathbb{Z}[x]$ of degree $k$ with positive leading coefficient, there are a constant $C_h \ge1$ and an integer $N_h \ge1$ such that, whenever $N \ge N_h$ and $A \subseteq[N]$ satisfy

one has

The exponent is common to all polynomials of a fixed degree. The constants $C_h$, $N_h$ may depend on every coefficient of $h$, and $h$ remains fixed as $N$ grows. The theorem concerns ordinary integer differences; zero values of $h$ impose no restriction on $A$.

History and relation to earlier work

Furstenberg [6] and Sárközy [14] proved that every set of positive upper density contains two elements whose difference is a nonzero square. Their proofs introduced distinct ergodic and Fourier-analytic approaches to this recurrence problem. Kamae and Mendès France [9] extended the positive-density conclusion to nonconstant intersective polynomials. Together with the elementary congruence obstruction above, this identifies the exact class of nonconstant polynomials for which the density of an avoiding set must tend to zero.

The quantitative problem has driven successive refinements of Fourier density-increment methods. For square differences, Pintz, Steiger, and Szemerédi [12] introduced a substantially stronger iteration. Bloom and Maynard [3] later obtained the bound

using estimates for the additive energy of rational numbers. For general intersective polynomials, Lucier [10] established quantitative bounds using auxiliary polynomials adapted to progressions, and Rice [13] developed stronger estimates by combining iteration with sieved polynomial sums. Arala [2] extended the Bloom–Maynard bound to every intersective polynomial of degree at least two, with a constant depending only on the degree in the exponent of $\log N$.

Green and Sawhney [7], Theorem 1.1 obtained the square-difference bound

Their arithmetic level-$d$ inequality and random sparsification also underpin the recent general-polynomial result of Adajar et al. [1], Theorem 1.1: for every fixed intersective $h$ of degree at least two and every fixed $0<\mu<1/2$,

for sufficiently large $N$. eq:1 gives a fixed power of $N$ as the saving, with an exponent common to all polynomials of each fixed degree.

The immediate predecessor for our method is the power-saving theorem for square differences in [11], Theorem 1.1. Its proof uses finite reflection-positive tuple laws and a signed multilinear functional. Hatami [8] gave a multilinear Hölder criterion for graph norms. Reflection and Cauchy–Schwarz arguments are central to the reflection-group constructions of Conlon and Lee [4]. Here we adapt the specific laws and functional of [11], with all required proofs included, and supply the polynomial and change-of-scale arguments described next.

The argument and its principal constructions

The square-difference method constructs a multilinear form which dominates a fixed power of density, and proves that this form contracts on passing to shorter intervals. We retain this organization, but polynomial values require two additional compatibilities: the polynomial changes when one passes to a progression, and the local distribution supplied by the multilinear construction need not be the natural distribution of polynomial values.

Normalizing the local coefficients. We first pass from the polynomial in Theorem 1.1 to a polynomial $g(x)=h(r+D_0x)/M$ for which the first nonzero Taylor coefficient at one chosen root in each $\mathbb{Z}_p$ is a unit. Only finitely many primes require the extra division by $M$. The multiplicity of each chosen root is retained, and the representative $r$ is chosen so that positive inputs for $g$ give positive inputs for $h$. The purpose is to make the local numerical choices that determine the final exponent depend only on the degree.

A family closed under passage to progressions. Section 2 retains the auxiliary construction for an arbitrary fixed intersective polynomial $h$. It gives integer polynomials $h_l$, indexed by $l\in\mathbb{N}_+$, with $h_1=h$, and a fixed completely multiplicative function $\lambda:\mathbb{N}_+\to\mathbb{N}_+$ such that, for all positive integers $l,D$,

Consequently, a subset of an arithmetic progression of step $\lambda(D)$ which avoids nonzero $h_l$-differences becomes, in progression coordinates, a set avoiding nonzero $h_{lD}$-differences. Compatible roots in the rings $\mathbb{Z}_p$ produce the family. Its coefficient bounds, complete exponential-sum estimates, and local lifting properties are uniform in $l$.

Applied to $g$, the family retains the zero valuation of the first nonzero coefficient at each chosen root: Lucier’s auxiliary formula preserves that valuation [10], Section 8. Thus every auxiliary polynomial for $g$ is nonconstant as an element of $\mathbb{F}_p[x]$ for every prime $p$. For $p > k$, its reduced derivative is nonzero, and the complete-sum estimates hold beyond a threshold depending only on $k$. At the finitely many smaller primes below that threshold, interpolation bounds the derivative valuation at some argument in terms of $k$ and $p$. These bounds make the lifting moduli and cycle length, and hence the number of tuple slots below, depend only on $k$.

Weighted forms that follow the family. For each sufficiently large prime $p$, Section 3 constructs a probability distribution on tuples $(z_v)_{v\in V} \in\mathbb{F}_p^V$, where $V$ is one fixed finite set of even cardinality $d$. A distinguished cycle of slots has increments among the values of $h_l$ at arguments where its derivative is nonzero modulo $p$, apart from a small exceptional part of the distribution. The constraint on an edge is weighted by the number of such arguments giving its increment. This weight has a Fourier bound uniform in the auxiliary polynomial and transforms exactly under the progression identity above.

The laws are reflection positive: the quadratic form obtained by splitting the slots across any of a specified family of reflections is positive semidefinite. They also have uniform individual marginals and a conditional mixing estimate. Section 4 derives the resulting signed Hölder inequality and estimates for Fourier lifts. At scale $N$, the lift retains the normalized Fourier coefficients of $1_A$ at rational frequencies whose denominators are at most a small fixed power of $N$. Integrating its product over the $d$ slots gives a nonnegative quantity $Y_l(N,A)$ satisfying

The associated seminorm permits comparison along progressions with a factor tending to one, rather than a fixed loss at every scale.

A kernel realizing the resulting increment distribution. The increment distribution of two slots is determined by the whole tuple construction. It is supported on regular polynomial values, but there is no reason for it to equal the distribution obtained by evaluating $h_l$ at a uniform argument. Section 5 constructs a signed kernel on actual positive values of $h_l$ whose Fourier transform approximates the desired local multipliers. At each sufficiently large prime, a density on regular arguments realizes the prescribed increment distribution. A truncated divisor expansion combines these densities, while uniform lifting handles characters of higher prime-power conductor. Minor arcs are controlled by elementary Weyl differencing.

Only support is needed to use this kernel: its bilinear form vanishes on two ordered subsets of an avoiding set, regardless of the signs of the kernel weights. Its Fourier approximation therefore gives cancellation for the chosen pair of slots. This separates the construction of a useful tuple law from the problem of realizing its pair distribution on integer polynomial values.

Contraction while the auxiliary parameter grows. Section 6 combines the kernel estimate with the seminorm comparison. A fixed positive proportion of assignments of residue classes at the finitely many small primes gives cancellation. The other assignments, together with the summable exceptional parts of the tuple laws, are bounded using shorter intervals and polynomials farther along the family. Section 7 closes the resulting recurrence with a bound that holds for some fixed $a,b>0$, simultaneously for every $l$ and every set $A$ avoiding nonzero $h_l$-differences:

The factor $l^b$ is necessary to accommodate the growth of the auxiliary polynomial: when $l$ is large relative to $N$, a uniform subpower bound suffices, while the kernel supplies contraction in the remaining range. For the normalized polynomial $g$, the parameters can be chosen so that $a/d$ depends only on $k$. Taking $l=1$ and using density domination bounds sets avoiding nonzero $g$-differences. Partitioning $A$ into residue classes modulo $M$, and using $Mg(n)=h(r+D_0n)$ with $r+D_0n\geq1$, then transfers the bound to $h$ and proves Theorem 1.1.

The polynomial argument thus adds exact covariance of the weighted tuple laws across the auxiliary family and a signed polynomial kernel realizing their pair multipliers. The initial normalization controls the structural parameters that determine the exponent. All the estimates needed from the reflection-positive method are proved below.

Notation and uniformity

We write $e(x)=\exp(2\pi i x)$ and use the negative sign in Fourier coefficients. For $\xi\in\mathbb{Q}/\mathbb{Z}$, let $q(\xi)$ be its reduced positive denominator, with $q(0)=1$. Averages, measures, and norms on finite sets use uniform probability measure unless a different measure is specified. For a positive integer $n$, write $\operatorname{rad}(n)=\prod_{p\mid n}p$, with $\operatorname{rad}(1)=1$. Empty products equal 1, and a product space with no coordinates is a singleton. Conditional expectations are with respect to the law specified in the surrounding statement.

The auxiliary-family development applies to any fixed intersective polynomial $h$ with chosen roots, and we write $k=\deg h$. Constants in $O(\cdot)$ and $\ll$ may depend on this fixed datum and on parameters already fixed from it; they are independent of the varying integers, sets, and auxiliary index $l$, unless a restriction is explicitly stated. Likewise, sufficiently-large thresholds are uniform in those varying data. We explicitly identify the structural choices that depend only on $k$ when the first nonzero coefficient at every chosen root is a unit. Bounds denoted $O_k(\cdot)$ are uniform over all fixed data of degree $k$ satisfying those choices. A bound $N^{o(1)}$ means $C_\varepsilon N^\varepsilon$ for every $\varepsilon>0$ under the same uniformity convention. All structural parameters are fixed before the asymptotic scale $N$ varies.

Auxiliary polynomials and local estimates

Passing to an arithmetic progression changes the polynomial whose values are forbidden differences. We first show that one such change can make the first nonzero Taylor coefficient at each chosen local root a unit. We then organize all subsequent changes into a coherent family $h_l$, indexed by positive integers, and show that this local property is preserved. The family and its local estimates remain available for an arbitrary fixed intersective polynomial; the unit condition will let us choose their structural parameters using only the degree. The estimates give decay of complete exponential sums at large primes, together with residue classes on which polynomial values lift uniformly and support cycles of one fixed length.

Throughout this section, $h\in\mathbb{Z}[x]$ is the fixed intersective polynomial of degree $k\geq2$, with positive leading coefficient $a$.

A family compatible with progressions

We use Lucier’s auxiliary-polynomial construction [10], Section 8, in which a compatible root at each prime determines the normalization by its multiplicity. We give the construction and its inheritance rule explicitly, before proving the uniform local estimates needed here.

Here $\mathbb{Z}_p$ denotes the ring of compatible residues modulo the powers of a prime $p$, and $v_p$ its valuation, with $v_p(0)=\infty$. Every nonzero element is a power of $p$ times a unit, so $\mathbb{Z}_p$ is an integral domain. Intersectivity supplies a root of $h$ modulo every power of $p$. These roots can be chosen compatibly: at each stage, retain a residue class having root extensions to arbitrarily high powers, and then choose one of its finitely many refinements with the same property. Fix the resulting root $\zeta_p\in\mathbb{Z}_p$ for each prime $p$, and write

Thus $m_p$ is the multiplicity of the chosen root.

The coefficient $b_{m_p,p}$ is a unit for all sufficiently large primes, with a threshold depending only on $h$. To see this, factor $h=c\prod_i g_i^{d_i}$, where $c$ is a nonzero integer and the $g_i$ are pairwise nonassociate primitive irreducible polynomials in $\mathbb{Z}[x]$. Cleared Bézout identities for $(g_i,g_j)$, $i\ne j$, and for $(g_i,g_i')$ have nonzero integer right-hand sides. Exactly one factor $g_i$ vanishes at $\zeta_p$, and $g_i'(\zeta_p)\ne0$. Outside the finite set of primes dividing these right-hand sides or $c$, the derivative and all other factor values at $\zeta_p$ are units. Hence $m_p=d_i$ and

is a unit. This argument allows repeated factors of $h$.

We will sometimes impose the following additional condition on the chosen roots: