theoretical computer scientists

Abstract: The arithmetic regularity lemma of Green is a fundamental tool in additive combinatorics, providing a decomposition of arbitrary subsets of finite Abelian groups into pseudorandom pieces with respect to Fourier analysis. Over the years, it has found numerous applications in combinatorics, property testing, and theoretical computer science. However, Green's proof is existential, and many algorithmic applications require an efficient procedure for constructing the underlying regular decomposition. De, Nadimpalli, and Servedio (SODA 2022) developed a query-efficient constructive regularity framework over $\mathbb{F}_2^n$ and explicitly raised the problem of extending their dimension-independent sumset-estimation algorithm to other finite Abelian groups. In this work, we resolve this question for finite Abelian groups of constant torsion. We develop the first constructive version of Green's {\em Arithmetic Regularity Lemma} in this general setting. Given oracle access to a subset of the group, our algorithm computes an implicit description of a regular decomposition together with an oracle representation of a structured subset satisfying the guarantees of Green's lemma. The algorithm runs in time and query complexity polynomial in the regularity parameters and the tower bound, while remaining independent of the size of the ambient group. As applications, we obtain efficient algorithms for removing linear configurations and approximately counting linear images, including weighted sumsets and suitable matrix systems of Cauchy--Schwarz complexity one, over finite Abelian groups of constant torsion. Our results provide a unified algorithmic framework extending dimension-independent constructive arithmetic regularity and its applications beyond $\mathbb{F}_2^n$.

Abstract: Let $S\subseteq \mathbb R^+$ be a subset of positive reals. We say that a metric space is an $S$-metric space if the range of the metric is a subset of $S\cup \{0\}$. In 2007, Delhommé, Laflamme, Pouzet, and Sauer characterized those sets $S$ for which the class of all finite $S$-metric spaces has the amalgamation property. Given an amalgamation class, a natural question is what its Ramsey expansion is. Ramsey expansions are known for classes of $S$-metric spaces for which $S\cup\{0\}$ carries a natural distance monoid structure, for example for $S=\mathbb Q^+$ or when $S$ is topologically closed. In this paper, we extend this result to all amalgamation classes of finite $S$-metric spaces, thereby completing the Ramsey-theoretic classification of amalgamation classes of finite $S$-metric spaces.