Simultaneous popular polynomial differences over finite fields

Abstract: 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=(ω1,\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 $\mathbb F_p<sup>n$ 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.

• The paper extends Green’s popular difference theorem by proving that for any collection of linearly independent polynomials, a fixed nonzero d exists that simultaneously guarantees near-random density for all subconfigurations in large prime fields.
• It employs an arithmetic regularity decomposition, polynomial Szemerédi theorem, and Fourier-analytic techniques to derive quantitative bounds and control exponential sums.
• The study also constructs counterexamples in vector spaces, sharply contrasting finite field behavior and revealing inherent limitations for simultaneous popularity in three-term progression configurations.

Simultaneous Popular Polynomial Differences Over Finite Fields

Introduction and Context

The study addresses the existence and limitations of simultaneous popular differences for polynomial configurations in dense subsets of finite fields. The classic backdrop is Green's popular difference theorem, which establishes that for any set $A \subseteq \mathbb{F}_p$ of density $\alpha$, there exists a nonzero $d$ such that the density of three-term arithmetic progressions $(x, x+d, x+2d)$ in $A$ with common difference $d$ is at least $\alpha^3 - \varepsilon$. This result can be seen as a density version of Khintchine-type recurrence theorems and connects with polynomial generalizations via the Bergelson-Leibman polynomial Szemerédi theorem.

The paper extends Green’s theorem by showing a multi-parameter generalization for polynomial patterns, answering the following: Given a family of linearly independent polynomials $\mathcal{P} = \{P_1, \dots, P_k\}$, can one find a nonzero $d$ such that every subconfiguration formed by the corresponding polynomial shifts is also present with near-random density? The work further establishes fundamental obstructions to such simultaneous phenomena in the vector space group setting.

Main Results

Positive Result: Simultaneous Popular Polynomial Differences

The principal positive theorem states that for any collection $\mathcal{P} \subseteq \mathbb{Z}[t]$ of $\alpha$0 linearly independent polynomials with $\alpha$1, every fixed density $\alpha$2 set $\alpha$3 (for all sufficiently large $\alpha$4) admits a nonzero $\alpha$5 such that:

$\alpha$6

for every choice of $\alpha$7 simultaneously. This unifies and strengthens earlier Khintchine- and Frantzikinakis-Kra-type recurrence phenomena: a fixed $\alpha$8 is simultaneously a popular difference for all possible subconfigurations of $\alpha$9.

Key Technical Ingredients

• Arithmetic Regularity Decomposition: Splits the indicator function into structured, pseudorandom, and negligible components, controlling their influence on polynomial counts. This approach leverages techniques from finite field additive combinatorics [BSST22], [T14].
• Peluse’s Polynomial Szemerédi Theorem: Supplies asymptotic independence for averages of the form $d$0, provided the polynomials are linearly independent [P19].
• Bohr Set Methods: Low-rank Bohr sets serve as approximate subgroups to control translation invariance even in the absence of nontrivial additive subgroups in $d$1 ([TV06]).
• Fourier-Analytic and Equidistribution Arguments: Quantitative bounds are obtained through careful estimation of exponential sums, in particular leveraging the Weil bound to control the distribution of polynomial values.

Strong Negative Result: Limitations for Simultaneous Progression Popularity

The paper provides a sharp limitation by constructing, for any fixed odd prime $d$2 and for sufficiently large $d$3, a set $d$4 of density $d$5 in which for every nonzero $d$6,

$d$7

for some absolute constant $d$8. Thus, it is impossible to enforce that both $d$9 and $(x, x+d, x+2d)$0 are simultaneously popular differences for three-term progressions across all sets in vector spaces over fixed finite fields, sharply contrasting the situation in cyclic groups of large prime order.

Notable Technical Innovation

• The construction exploits quadratic Fourier analysis and functional-analytic lifting between the torus $(x, x+d, x+2d)$1 and $(x, x+d, x+2d)$2, combining trigonometric polynomial approximation with combinatorial set construction in $(x, x+d, x+2d)$3.
• The method shows that interference between different subprogressions (e.g., by mixing $(x, x+d, x+2d)$4 and $(x, x+d, x+2d)$5) can always force at least one progression type to exhibit density lower than the random bound.

Implications

Theoretical Impact

• The positive theorem yields a broad class of simultaneous popular difference results for polynomial patterns in prime fields, expanding the reach of polynomial multiple recurrence in additive combinatorics.
• The negative result delineates a sharp dichotomy between the behavior of $(x, x+d, x+2d)$6 and the vector space setting $(x, x+d, x+2d)$7, providing a template for understanding the obstructions to simultaneous density phenomena for higher-dimensional or longer arithmetic progressions.
• These findings underscore the delicate structure of popularity phenomena—Green-style theorems are fragile under even moderate generalization to vector spaces.

Practical Consequences

• The polynomial generalizations may influence the design of pseudorandom objects, expanders, and uniformity testing algorithms where control over higher-order correlations is required.
• The limitations in the vector space setting suggest that one must be cautious when extending combinatorial density results to higher dimensions, motivating the search for more refined structural parameters or additional invariants.

Connections and Future Directions

• The results prompt new questions about the quantitative bounds for minimal field size $(x, x+d, x+2d)$8 required versus the degree and number of polynomials in $(x, x+d, x+2d)$9, given that known proofs (via regularity methods) impose tower-type bounds [FP21, FPZ23].
• The paper alludes to the possibility of further extending positive results to rational function configurations, with recent companion results providing analogues when higher degrees of independence and equidistribution are assured [HL25].
• Open problems remain regarding simultaneous popularity for mixed polynomial and progression-related configurations (for instance, combinations of three- and four-term progressions), and for nonlinear patterns in vector spaces.

Conclusion

This paper establishes a precise boundary for simultaneous popular difference phenomena in polynomial configurations within finite fields. The main positive result demonstrates that for any linearly independent polynomial collection, popular differences enforcing random-like statistical regularity exist simultaneously for all subconfigurations in large prime fields. Conversely, the negative result shows fundamental obstructions to such phenomena in vector spaces, even for modest configurations. The methodological innovations tightly combine ergodic-theoretic, Fourier-analytic, and combinatorial tools, enriching our foundational understanding of structure and randomness in finite field settings (2607.10051).