Large subsets of Local Fields not containing Configurations

Abstract

For certain families of functions $\{f_q\}$ mapping $K^{n{v}_q}\to K^m$, where $K$ is a complete, nonarchimedean local field, we find a set $E$ of large Hausdorff dimension with the property that $f_q(x_1,\dots ,x_{v_q})$ is nonzero for any distinct points $x_1,\dots ,x_{v_q}\in E$. 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.

17 Pages

References in corpus (1)

• Sets of large dimension not containing polynomial configurations