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math.CO2026

Spectral Incidence Bounds over Finite Fields

Yao Zhi

Let $\Fq$ be a finite field, let $F,G\in\Fq[T]$ be nonconstant, and let $J\in\Fq[T]$. We study the parameterized two-parameter family \[ C_{a,b}=\{(x,t)\in\Fq^2:t=J(x)+aF(x)+bG(x)+…

math.CO2026

An Improved Exponent for Products of Differences over Arbitrary Finite Fields

Yao Zhi

Murphy and Petridis proved that, for subsets of an arbitrary finite field $\F_q$, the condition $\abs A>q^{2/3-δ}$ implies $\abs{(A-A)(A-A)}>q/2$ whenever and i…

math.CO2026

Branch-Sensitive Density Increments for -Configurations

Yao Zhi

We refine the density-increment argument for -configurations developed by Beker. The point is not to introduce a new inverse theorem, but to retain information that is lost when…

math.CO2026

Exact Diameter Windows for Random Cayley Graphs on Odd-Order Abelian Groups

Yao Zhi

Let \(d\ge2\) be fixed and let \(G_n\) be finite abelian groups of odd orders \(N_n\to\infty\). We determine the centered diameter-\(d\) critical window for the standard random Cay…

math.CO2026

Quadratic Expansion over Prime Fields via Centered Collisions and Popular-Sum Amplification

Zhi Yao

Let be an odd prime, let have cardinality , and let be a non-degenerate quadratic polynomial. Writing $S=|A+A|…