Large subsets of Local Fields not containing Configurations

Robert Fraser

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arXiv · 1612.08231

Large subsets of Local Fields not containing Configurations

Robert Fraser

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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.

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Robert Fraser. 2016-12-25. Large subsets of Local Fields not containing Configurations. https://arxiv.org/abs/1612.08231

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