A counterexample to a conjecture of Sárközy on sums and products modulo a prime

Abstract: Let $p$ be a prime and, for $A\subseteq \mathbb F_p$, define $A^\ast=(A+A)\cup(AA)$. Sárközy conjectured that there exist constants $c>0$ and $p_0$ such that, for every prime $p>p_0$, every set $A\subseteq \mathbb F_p$ with $|A|>\left(\frac12-c\right)p$ satisfies $\mathbb F_p^\times\subseteq A^\ast$. We disprove this conjecture: for every odd prime $p\ge 5$, there exists a set $A\subseteq \mathbb F_p$ with $|A|=\frac{p-1}{2}$ such that $1\notin A^\ast$. Thus no positive constant $c$ can satisfy Sárközy's conjecture. Conversely, if $|A|>\frac{p}{2}$, then $A+A=\mathbb F_p$. Therefore the sharp threshold is exactly $\frac12$.

• The paper disproves Sárközy's conjecture by establishing that a set A with |A| > p/2 is necessary to cover all nonzero residues in Fₚ via A+A.
• The work employs combinatorial methods and graph decomposition, constructing explicit counterexamples through adjacency relations based on sum and product constraints.
• The analysis utilizes the pigeonhole principle and involutive transformations to reveal a sharp threshold, offering new insights for extremal problems in finite field additive combinatorics.

Disproving Sárközy's Conjecture on Sums and Products Modulo a Prime

Introduction

This work systematically addresses a conjecture posed by Sárközy in 2001 regarding the coverage of nonzero residues in the finite field $\mathbb{F}_p$ by the set $A^* = (A + A) \cup (AA)$, where $A \subseteq \mathbb{F}_p$. Sárközy conjectured the existence of constants $c>0$ and $p_0$ such that, for any prime $p > p_0$ and any set $A$ with $|A| > (\frac{1}{2} - c)p$, the set $A^*$ would cover all nonzero elements of the field, i.e., $\mathbb{F}_p^\times \subseteq A^*$. The paper presents a definitive negative resolution of this conjecture—establishing that the threshold is sharp at $A^* = (A + A) \cup (AA)$0, and there does not exist a positive $A^* = (A + A) \cup (AA)$1 for which the conjecture holds.

Main Results

Sharp Threshold for Complete Coverage

For all primes $A^* = (A + A) \cup (AA)$2, it is shown that if $A^* = (A + A) \cup (AA)$3, then the sumset $A^* = (A + A) \cup (AA)$4 already covers the entirety of $A^* = (A + A) \cup (AA)$5. The argument is based on double counting and the pigeonhole principle: for any $A^* = (A + A) \cup (AA)$6, the sets $A^* = (A + A) \cup (AA)$7 and $A^* = (A + A) \cup (AA)$8 are both larger than $A^* = (A + A) \cup (AA)$9, thus must intersect, ensuring $A \subseteq \mathbb{F}_p$0.

Explicit Counterexample Construction

The principal result of the paper is the construction, for every odd prime $A \subseteq \mathbb{F}_p$1, of $A \subseteq \mathbb{F}_p$2 with $A \subseteq \mathbb{F}_p$3 such that $A \subseteq \mathbb{F}_p$4.

Methodology of Construction

• The forbidden sums and products generating $A \subseteq \mathbb{F}_p$5 are encoded as adjacency relations on $A \subseteq \mathbb{F}_p$6, a graph with vertex set $A \subseteq \mathbb{F}_p$7. Edges exist if $A \subseteq \mathbb{F}_p$8 or $A \subseteq \mathbb{F}_p$9; loops occur at $c>0$0 if $c>0$1 or $c>0$2.
• The structure of $c>0$3 is decomposed into:
  • Two small exceptional components: $c>0$4 and $c>0$5
  • Possibly a two-vertex component from roots of $c>0$6
  • Multiple $c>0$7-cycles, each corresponding to the cross-ratio (anharmonic) orbit generated by the involutions $c>0$8 and $c>0$9
• An independent set is constructed by selecting from each component: $p_0$0 from $p_0$1; $p_0$2 from $p_0$3; one arbitrary root if present; and three alternating vertices in each $p_0$4-cycle. This achieves a total size of $p_0$5.

Optimality and Extremal Value

It is rigorously established that $p_0$6 is both a lower and upper bound for the largest $p_0$7 for which $p_0$8. If $p_0$9, $p > p_0$0 covers all of $p > p_0$1 by the earlier argument.

Implications

Theoretical Consequences

• Resolution of Conjecture: The result conclusively disproves Sárközy's conjecture by exhibiting, for all sufficiently large primes, sets larger than $p > p_0$2 for any $p > p_0$3 that nonetheless do not satisfy the conjecture's coverage requirement.
• Threshold Tightness: The sharp threshold at $p > p_0$4 aligns with classical sumset phenomena in finite fields, emphasizing the fundamental barrier created by simple combinatorial constraints in covering residue classes via additive and multiplicative operations.
• Connections to Finite Field Arithmetic: The analysis is motivated by classical Diophantine, combinatorial, and Ramsey-type questions in finite fields—specifically, density thresholds for the solvability of equations with restricted solution sets. The independent set construction via involutive actions demonstrates the deep interplay between algebraic transformations and combinatorial configurations.

Practical and Methodological Insights

• The methodology underlying the construction—analyzing adjacency graphs induced by sum and product constraints—is general and could inform further extremal problems involving other algebraic relations over finite fields.
• The formalization of the main statements using Lean 4 and automated theorem provers such as Aristotle showcases the feasibility and rigor of using interactive proof assistants in advanced additive combinatorics and extremal set theory.

Prospects for Further Research

• Generalizations: One direction is to consider analogous sum-product conjectures in larger finite fields, product of fields, or rings, and for other forms of algebraic relations.
• Algorithmic Implications: The constructive approach and graph-theoretic analysis presented might have algorithmic consequences in efficiently finding independent sets with extremal properties in other algebraic combinatorial structures.
• Connections to Additive Combinatorics: The sharp dichotomy observed here may inform investigations into analogues of the sum-product phenomenon in more general settings, as well as stimulate renewed classification of extremal sets avoiding certain algebraic configurations.
• Automated Theorem Proving: The application of interactive and automated theorem provers for such combinatorial extremal problems can be expanded to other deep conjectures in combinatorics and finite fields, aiding both understanding and verification.

Conclusion

This paper demonstrates that the threshold for ensuring that $p > p_0$5 covers all nonzero residues in $p > p_0$6 is exactly $p > p_0$7, invalidating Sárközy's conjecture that a smaller threshold suffices. The construction hinges on a precise combinatorial analysis of sum and product constraints via involutive transformations, yielding an independent set maximum of $p > p_0$8, and no greater. This result sharpens our understanding of arithmetic combinatorics in finite fields and opens avenues for both generalization and formal verification in the domain.