Intersective Polynomial Difference-Free Sets Receive Stronger Density Bounds

The Furstenberg–Sárközy problem asks how large a subset of $[1,X]$ can be if none of its nonzero differences lies in a prescribed polynomial sequence. For square differences, Green and Sawhney recently obtained a quasipolynomial quantitative bound using an arithmetic level-$d$ inequality and a strengthened density-increment scheme. This paper extends that approach to every intersective polynomial $h\in\mathbb{Z}[x]$, meaning a polynomial with a root modulo every positive integer.

The authors prove that, for every $0<\mu<1/2$, there are positive constants $c_0$ and $X_{\min}$, depending on $h$ and $\mu$, such that

for $X>X_{\min}$. Here $D(h(\mathbb{N}),X)$ denotes the largest size of a set $A\subseteq[1,X]$ whose difference set contains no nonzero value in $h(\mathbb{N})$. The result is a quantitative extension from square differences to general intersective polynomial differences.

Core Contribution

The central contribution is an adaptation of the arithmetic level-$d$ method to polynomial differences. The proof combines a density-increment framework with new exponential-sum estimates developed using sieve methods in the style of Rice. It also uses the random sparsification procedure of Green and Sawhney to retain the structure needed for the iteration.

The argument accounts for the local arithmetic constraints associated with intersective polynomials. This is essential because intersectivity is expressed through the existence of roots modulo every positive integer, and those local conditions shape the polynomial-difference problem across moduli.

Technical Approach

The paper develops exponential-sum estimates suitable for the polynomial setting and incorporates them into an arithmetic level-$d$ density-increment strategy. The sieve component supplies estimates for polynomial values, while random sparsification helps manage the distributional structure required by the iteration.

Together, these ingredients extend the improved square-difference method to general intersective polynomials. Rather than relying only on a qualitative recurrence statement, the proof produces an explicit asymptotic form for the extremal quantity $D(h(\mathbb{N}),X)$.

Results and Analysis

The theorem reaches every exponent $\mu<1/2$. The authors describe this as the best quantitative upper bound currently known for sets avoiding intersective polynomial differences, improving on prior work of Arala.

The evidence is theoretical: the paper proves an extremal upper bound and does not present numerical experiments or empirical benchmarks. Its significance is therefore in the quantitative improvement and in the extension of arithmetic level-$d$ methods from squares to a broad class of polynomial difference sets.

Scope and Caveats

The theorem applies only to intersective polynomials, so it does not cover polynomials that fail to have roots modulo some positive integer. The constants $c_0$ and $X_{\min}$ depend on both $h$ and $\mu$, and the conclusion is asymptotic, applying for $X>X_{\min}$. The result is an upper bound on extremal set size; the supplied theorem statement does not provide matching lower bounds.