On the Lebesgue measure of one generalised set of subsums of geometric series

TL;DR

This paper investigates the Lebesgue measure of a generalized set of subsums of a geometric series, extending previous results and addressing a broader, less restrictive class of such sets.

Contribution

It advances the understanding of the Lebesgue measure for generalized subsum sets beyond the homogeneous case, filling a gap in the existing literature.

Findings

Confirmed the positivity of Lebesgue measure for a broader class of subsum sets

Extended previous results to less restrictive, more general geometric series

Provided new insights into the measure-theoretic properties of these sets

Abstract

In the present paper, we study a set that can be treated as a generalised set of subsums for a geometric series. This object was discovered independently in various mathematical aspects. For instance, it is closely related to various systems of representation of real numbers. The main object of this paper was particularly studied by R. Kenyon, who brought up a question about the Lebesgue measure of the set and conjectured that it is positive. Further, Z. Nitecki confirmed the hypothesis by using nontrivial topological techniques. However, the aforementioned result is quite limited, as this particular case should satisfy a rigid condition of homogeneity. Despite the limited progress, the problem remained understudied in a general framework.