Finite Field Analogue of Verstraëte's Conjecture

• The paper introduces a sharp asymptotic formula for F_k(q;h) using combinatorial methods and character sum arguments.
• It partitions Fq* into cosets and employs Weil’s bound and combinatorial lemmas to control the avoidance of k-term products.
• The results reveal a dichotomy between linear (Θ(q)) and sublinear (O(√q)) growth, fully characterizing the finite field analogue of classic extremal product-set problems.

The finite field analogue of Verstraëte’s conjecture concerns the maximal size of subsets of $F_q^*$ that avoid $k$-term products lying in the value set of a polynomial $h(x)\in F_q[x]$. This question generalizes classical extremal product-set problems over the integers, recasting them into the arithmetic and combinatorial framework of finite fields. The problem admits a dichotomous solution governed by algebraic invariants of $h$ and $k$, with precise asymptotics characterized in recent work by Lee–Yip–Yoo (Lee et al., 23 Jan 2026).

1. Formulation in Finite Fields

Let $q$ denote a prime power and $F_q$ the finite field of order $q$; $F_q^*$ is its multiplicative group. Fix an integer $k\ge2$ and a nonconstant $k$0. The central quantity is

$k$1

i.e., $k$2 avoids $k$3-products in the value set $k$4. This generalizes integer analogues studied by Erdős, Sárközy, Sós, and others, in which one avoids products that are perfect squares or more generally, elements in $k$5.

2. Dichotomy and Asymptotics

In analogy with Verstraëte’s conjecture over $k$6—which posited that the corresponding extremal size for integers grows either linearly or like the counting function of squares—the finite field setting yields a dichotomy for $k$7: as $k$8,

• $k$9, or
• $h(x)\in F_q[x]$0,

with the exact threshold determined by modular invariants attached to $h(x)\in F_q[x]$1 and $h(x)\in F_q[x]$2. The critical combinatorial parameter is $h(x)\in F_q[x]$3, counting cosets of suitable subgroups which avoid certain sumset structure.

3. Main Theorem and Characterization

Let $h(x)\in F_q[x]$4 be a decomposition in $h(x)\in F_q[x]$5, where $h(x)\in F_q[x]$6 is maximized and $h(x)\in F_q[x]$7 is not a perfect power. Define $h(x)\in F_q[x]$8. Let $h(x)\in F_q[x]$9 be the unique subgroup of $h$0 of index $h$1, and let $h$2 be a generator of $h$3 writing $h$4 for unique $h$5. The crucial combinatorial maximum is

$h$6

where $h$7.

The principal result is

$h$8

where the $h$9 term depends on $k$0 and $k$1 (Lee et al., 23 Jan 2026). The regime is linear in $k$2 if $k$3, and $k$4 otherwise. For $k$5, an explicit formula holds:

$k$6

4. Proof Strategies and Construction

Proofs consist of matching upper bounds and explicit constructions:

Upper Bound:\ Partition $k$7 into cosets of $k$8: $k$9, $q$0. For large cosets ($q$1 big), a character sum argument (Weil’s bound and combinatorial lemma extending Gyarmati’s approach) establishes that if $q$2 (with $q$3 large$q$4), then forbidden $q$5-products exist, contradicting the avoidance property. Thus, $q$6 and $q$7.

Construction:\ Select $q$8 of size $q$9 with $F_q$0; set $F_q$1. Then for any $F_q$2 distinct elements from $F_q$3, the product lies in $F_q$4 for $F_q$5, thus not in $F_q$6.

Auxiliary and Character Estimates:\ Key character sums utilize Weil’s bound: For nontrivial multiplicative character $F_q$7 of order $F_q$8 and $F_q$9 not a $q$0th power,

$q$1

This, along with Cauchy–Schwarz, underpins the analogues of Gyarmati’s lemma that force forbidden products in large sets.

5. Structural Conditions and Integer Comparison

Maximal $q$2 in $q$3 ensures $q$4 is not a nontrivial $q$5th power, enforcing applicability of Weil’s bound. When $q$6, $q$7, bringing a dichotomy between linear and sublinear extremal set sizes. If $q$8 is square-free ($q$9), $F_q^*$0 and $F_q^*$1, enforcing the $F_q^*$2 bound.

Contrasting with integers, the finite field scenario yields full solutions due to the regularity of coset decomposition. Over $F_q^*$3, Verstraëte’s conjecture is generally unresolved and fails for some $F_q^*$4 (e.g., $F_q^*$5).

6. Corollaries, Extremal Configurations, and Open Problems

• Explicit evaluations of $F_q^*$6 in the case $F_q^*$7.
• Near-maximal sets $F_q^*$8 are unions of $F_q^*$9 cosets of $k\ge2$0, up to $k\ge2$1 error.
• Both regime types are witnessed:
  • For $k\ge2$2, $k\ge2$3 nonsquare, and $k\ge2$4, $k\ge2$5, so only $k\ge2$6 size is achieved.
  • For $k\ge2$7 and $k\ge2$8, $k\ge2$9 of size $k$00 exists avoiding $k$01-products in $k$02; conjecturally, this is optimal.
• Open directions include:
  • Determining for which $k$03 pairs one can realize intermediate exponents $k$04 with $k$05 (with $k$06 conjectured optimal for $k$07, $k$08),
  • Refining the $k$09 error and transitions when $k$10,
  • Extending to rational functions or more general algebraic images.

In essence, Lee–Yip–Yoo (Lee et al., 23 Jan 2026) have established that for fixed $k$11 and $k$12, the finite field analogue of Verstraëte's conjecture is governed by the combinatorial invariant $k$13, dictating the size of product-avoiding sets with a sharp dichotomy and complete asymptotic description.