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 $h\in F_q[x]$. The central quantity is

$$F_k(q;h) = \max\{\, |A|: A\subseteq F_q^*\text{ and } a_1a_2\cdots a_k \ne h(x),\ \forall\,\text{distinct}\ a_i\in A, \forall\, x\in F_q \,\},$$

i.e., $A$ avoids $k$-products in the value set $h(F_q)$. 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 $h(\mathbb{Z})$.

2. Dichotomy and Asymptotics

In analogy with Verstraëte’s conjecture over $\mathbb{Z}$—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 $F_k(q;h)$: as $q\to\infty$,

• $F_k(q;h)=\Theta(q)$, or
• $F_k(q;h)=O(\sqrt{q})$,

with the exact threshold determined by modular invariants attached to $h$ and $k$. The critical combinatorial parameter is $m(k,n;s)$, counting cosets of suitable subgroups which avoid certain sumset structure.

3. Main Theorem and Characterization

Let $h(x)=C\cdot f(x)^{\ell}$ be a decomposition in $F_q[x]$, where $\ell$ is maximized and $f$ is not a perfect power. Define $n := \gcd(\ell,q-1)$. Let $H$ be the unique subgroup of $F_q^*$ of index $n$, and let $g$ be a generator of $F_q^*$ writing $C\in g^sH$ for unique $s\in\{0,\dots,n-1\}$. The crucial combinatorial maximum is

$$m(k,n;s) := \max \bigl\{|B|: B\subseteq\mathbb{Z}/n\mathbb{Z}\text{ and } s\not\in k\cdot B\bigr\},$$

where $k\cdot B = \{b_1+\cdots+b_k \bmod n: b_i\in B\}$.

The principal result is

$F_k(q;h) = \frac{m(k,n;s)}{n} q + O(\sqrt{q}),$

where the $O(\sqrt{q})$ term depends on $k$ and $\deg h$ (Lee et al., 23 Jan 2026). The regime is linear in $q$ if $m(k,n;s)>0$, and $O(\sqrt{q})$ otherwise. For $\gcd(k,n)=1$, an explicit formula holds:

$$m(k,n;s) = \max_{d|n} \bigl(\lfloor \frac{d-2}{k} \rfloor+1\bigr)\cdot\frac{n}{d}.$$

4. Proof Strategies and Construction

Proofs consist of matching upper bounds and explicit constructions:

Upper Bound:\ Partition $A$ into cosets of $H$: $A=\bigcup_{i=0}^{n-1}A_i$, $A_i\subset g^i H$. For large cosets ($|A_i|$ big), a character sum argument (Weil’s bound and combinatorial lemma extending Gyarmati’s approach) establishes that if $s\in k\cdot B$ (with $B=\{i: |A_i|$ large$\}$), then forbidden $k$-products exist, contradicting the avoidance property. Thus, $|B|\le m(k,n;s)$ and $|A|\le \frac{m(k,n;s)}{n}q+O(\sqrt{q})$.

Construction:\ Select $B_0\subset \mathbb{Z}/n\mathbb{Z}$ of size $m(k,n;s)$ with $s\notin k\cdot B_0$; set $A_0 = \bigcup_{i\in B_0}g^iH$. Then for any $k$ distinct elements from $A_0$, the product lies in $g^t H$ for $t\in k\cdot B_0\ne s$, thus not in $h(F_q)=g^s H$.

Auxiliary and Character Estimates:\ Key character sums utilize Weil’s bound: For nontrivial multiplicative character $\chi$ of order $d > 1$ and $f$ not a $d$th power,

$$|\sum_{x\in F_q} \chi(f(x))|\le (\deg f - 1)\sqrt{q}.$$

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 $\ell$ in $h(x)=C f(x)^\ell$ ensures $f$ is not a nontrivial $d$th power, enforcing applicability of Weil’s bound. When $\ell>1$, $n=\gcd(\ell,q-1)>1$, bringing a dichotomy between linear and sublinear extremal set sizes. If $h$ is square-free ($\ell=1$), $n=1$ and $m(k,1;0)=0$, enforcing the $\sqrt{q}$ bound.

Contrasting with integers, the finite field scenario yields full solutions due to the regularity of coset decomposition. Over $\mathbb{Z}$, Verstraëte’s conjecture is generally unresolved and fails for some $h$ (e.g., $h(x)=x^3$).

6. Corollaries, Extremal Configurations, and Open Problems

• Explicit evaluations of $m(k,n;s)$ in the case $\gcd(k,n)=1$.
• Near-maximal sets $A$ are unions of $m(k,n;s)$ cosets of $H$, up to $O(\sqrt{q})$ error.
• Both regime types are witnessed:
  • For $h(x)=\alpha x^2-1$, $\alpha$ nonsquare, and $q=p^2$, $n=1$, so only $O(\sqrt{q})$ size is achieved.
  • For $h(x)=\alpha x^m-1$ and $q=p^m$, $A\subseteq F_p^*$ of size $\Theta(p)=\Theta(q^{1/m})$ exists avoiding $k$-products in $h(F_q)$; conjecturally, this is optimal.
• Open directions include:
  • Determining for which $(h,k,q)$ pairs one can realize intermediate exponents $q^\theta$ with $0<\theta<1/2$ (with $\theta=1/m$ conjectured optimal for $h(x)=\alpha x^m-1$, $\alpha\not\in (F_q^*)^m$),
  • Refining the $O(\sqrt{q})$ error and transitions when $m(k,n;s)=0$,
  • 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 $h$ and $k$, the finite field analogue of Verstraëte's conjecture is governed by the combinatorial invariant $m(k,n;s)$, dictating the size of product-avoiding sets with a sharp dichotomy and complete asymptotic description.