Simultaneous popular polynomial differences over finite fields
Green's popular difference theorem says that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(α\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x)1_A(x+d)1_A(x+2d) \geq α^3-\varepsilon. \] We show that a stronger simultaneous popular difference phenomenon holds for polynomial configurations. Namely, if $\mathcal P=\{P_1,\dots,P_k\} \subset \mathbb Z[t]$ is a fixed collection of linearly independent polynomials with zero constant terms, we show that for every \(\varepsilon>0\), all sufficiently large primes \(p\), and every set \(A\subseteq\mathbb F_p\) of density \(α\), there exists a nonzero \(d\in\mathbb F_p\) such that \[ \mathbb E_{x\in\mathbb F_p} 1_A(x) \prod_{i=1}^k 1_A\bigl(x+P_i(d)\bigr)^{ω_i} \geq α^{1+\sum_iω_i}-\varepsilon \] simultaneously for every \(ω=(ω_1,\dots,ω_k)\in\{0,1\}^k\). We also show that such simultaneous popular difference phenomena have sharp limitations by proving that for every sufficiently large prime \(p\), there is a constant \(c>0\) such that, for all sufficiently large \(n\), one can find a set \(A\subseteq\mathbb F_p^n\) of density \(1/2+o_n(1)\) satisfying \[ \max_{d\neq 0} \min\left\{ \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+d)1_A(x+2d), \mathbb E_{x\in\mathbb F_p^n} 1_A(x)1_A(x+2d)1_A(x+4d) \right\} \leq \frac18-c. \] That is, the strengthening of Green's result, in this case over $\mathbb F_p^n$ for $p$ fixed and $n$ tending to infinity, requiring that both \(d\) and \(2d\) are simultaneously popular differences for three-term arithmetic progressions is false.
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Turtwig · 2026-07-20 12:27:07 EST
Summary:
This paper extends Green's popular difference theorem to polynomial configurations over finite fields, showing that for a collection of linearly independent polynomials with zero constant terms, there exists a single nonzero $d$ such that all subconfigurations of the polynomial pattern occur with essentially random density. The paper also demonstrates that this simultaneous popular difference phenomenon has sharp limitations by constructing sets in $\mathbb{F}_p^n$ where both $d$ and $2d$ cannot simultaneously be popular differences for three-term arithmetic progressions.
Mathematical/empirical assessment:
The paper provides a clear and rigorous proof of the main theorem using techniques from additive combinatorics, including arithmetic regularity decomposition and Fourier analysis. The key result is supported by detailed lemmas and bounds, particularly leveraging the Weil bound for exponential sums and properties of Bohr sets. The second theorem is also well-supported, with a construction based on quadratic forms and Fourier analysis that effectively demonstrates the limitations of simultaneous popular differences.
Strengths:
- The paper presents a novel extension of Green's theorem to polynomial configurations, which is a significant contribution to additive combinatorics.
- The proofs are thorough and well-structured, with clear use of known results and techniques from the field.
- The counterexample in the second theorem is carefully constructed and logically sound, demonstrating the limitations of the simultaneous popular difference phenomenon.
Concerns:
- The paper assumes familiarity with advanced topics in additive combinatorics, which may make it less accessible to readers without a strong background in the area.
- Some technical details, such as the use of specific lemmas and the handling of Fourier coefficients, could benefit from more explicit explanation or motivation.
Final decision: Strong accept