Large subsets of Local Fields not containing Configurations

TL;DR

This paper constructs large Hausdorff dimension subsets within local fields that avoid certain algebraic configurations, including 3-term arithmetic progressions, demonstrating rich combinatorial structures in nonarchimedean settings.

Contribution

It introduces a method to find large subsets of local fields avoiding specified polynomial configurations, extending combinatorial results to nonarchimedean local fields.

Findings

Existence of large Hausdorff dimension sets avoiding polynomial configurations

Construction of subsets in local fields without 3-term arithmetic progressions

Application to the ring of integers of local fields

Abstract

For certain families of functions $\{f_q\}$ mapping $K^{nv_q} \to K^m$, where $K$ is a complete, nonarchimedean local field, we find a set $E$ of large Hausdorff dimension with the property that $f_q(x_1, \ldots, x_{v_q})$ is nonzero for any distinct points $x_1, \ldots, x_{v_q} \in E$. In particular, this result can be applied to show that the ring of integers of any local field contains a subset of Hausdorff dimension $1$ not containing any nondegenerate 3-term arithmetic progressions.