Missing digits points near manifolds

TL;DR

This paper studies the distribution of points with missing digits near manifolds, showing they are evenly spread and achieve optimal dimension reduction, with results on distance sets and product sets containing intervals.

Contribution

It establishes new distribution and intersection properties of missing digits sets near manifolds, including dimension reduction and interval containment results.

Findings

Missing digits points distribute evenly around non-degenerate submanifolds.

Intersecting missing digits sets with submanifolds achieves optimal dimension reduction.

Pinned distance sets contain non-trivial intervals regardless of the pin location.

Abstract

We consider a problem concerning the distribution of points with missing digits coordinates that are close to non-degenerate analytic submanifolds. We show that large enough (to be specified in the paper) sets of points with missing digits coordinates distribute 'equally' around non-degenerate submanifolds. As a consequence, we show that intersecting those missing digits sets with non-degenerate submanifolds always achieve the optimal dimension reduction. On the other hand, we also prove that there is no lack of points with missing digits that are contained in non-degenerate submanifolds. Among the other results, 1. we prove that the pinned distance sets of those missing digits sets contain non-trivial intervals regardless of where the pin is. 2. we prove that for each $\epsilon>0,$ for missing digits sets $K$ with large bases, simple digit sets (to be specified in the paper), and $\dim_{H} K>3/4+\epsilon,$ the arithmetic product sets $K\cdot K$ contains non-trivial intervals.