arXiv subjects

Fred Tyrrell

Publications and source records attributed to Fred Tyrrell.

4 recordsLinked to original sources

arXiv2026

Multiplicative Subgroups of Prime Fields Are Not Sumsets

Let $H \leq \mathbb{F}_p^*$ be a proper multiplicative subgroup, and suppose that $H = A+B$ for some $A,B \subseteq \mathbb{F}_p$. We prove that either one of the summands is a singleton, or $|A|=|B|=2$ and $|H|=4$. In particular, no proper multiplicative subgroup of $\mathbb{F}_p^*$ can be written as $A+B$ with $|A|,|B|>2$. Our proof builds on the Hanson-Petridis polynomial method and Kalmynin's subsequent resolution of S\'ark\"ozy's conjecture for quadratic residues. Using Kalmynin's $|A|=|B|$ theorem as a structural input, we develop uniform combinatorial and arithmetic arguments which apply to multiplicative subgroups of arbitrary index.

math.CO↗

arXiv2026

Beating Product Constructions for Linear Equations Over Finite Fields

We show that for any $A\subseteq \mathbb{F}_q^n$ lacking non-trivial solutions to a translation-invariant linear equation of genus one, meaning that no nonempty proper subset of the coefficients sums to $0$, there is a set $B\subseteq \mathbb{F}_q^m$ in some higher dimension which also lacks non-trivial solutions, such that \[|B|^{1/m}>|A|^{1/n}.\] In particular, this implies that no fixed cap set in $\mathbb{F}_3^n$ gives an asymptotically optimal lower bound by direct products alone.

math.CO↗

arXiv2025

Bounded Exponential Sums with Multiplicative Coefficients

We investigate when the exponential sum $S_f(x,\alpha) := \sum_{n\le x}f(n)\mathrm{e}(n\alpha)$ is bounded, for a multiplicative function $f$ and $\alpha\in\mathbb{R}$. We show that under natural assumptions, $S_f(x,\alpha)$ is bounded only when $f$ is very close to a twisted Dirichlet character $\chi(n)n^{it}$. We obtain sharper classification results for functions that are completely multiplicative or take only finitely many values, including a complete classification in the case when $f$ is completely multiplicative and $\alpha$ is irrational. We also prove a stronger classification under the assumption that the sum is bounded for a positive measure set of $\alpha$.

math.NT↗

arXiv2022

New Lower Bounds for Cap Sets

A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x+y+z=0$ other than when $x=y=z$. In this paper, we provide a new lower bound on the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$.

math.CO↗