Exceptional Points for Density Modulo 1

TL;DR

This paper investigates the size and structure of the exceptional set of points where dilations of diverging sequences are not dense modulo 1, revealing that their Hausdorff and box dimensions can vary continuously between 0 and 1.

Contribution

It demonstrates that the dimension of the exceptional set can take any value between 0 and 1 and analyzes dimensions of related subsets with specific binary expansion properties.

Findings

The dimension of the exceptional set can be any value between 0 and 1.

Dimensions of certain subsets with binary constraints are explicitly calculated.

Results extend understanding of the size and complexity of non-dense dilations modulo 1.

Abstract

It is well known that almost every dilation of a sequence of real numbers, that diverges to $\infty$, is dense modulo~1. This paper studies the exceptional set of points -- those for which the dilation is not dense. Specifically, we consider the Hausdorff and modified box dimensions of the set of exceptional points. In particular, we show that the dimension of this set may be any number between 0 and 1. Similar results are obtained for two ``natural'' subsets of the set of exceptional points. Furthermore, the paper calculates the dimension of several sets of points, defined by certain constraints on their binary expansion.