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

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.