Jan Šustek

On the Lebesgue measure of the expressible set of certain sequences

Abstract The paper gives a condition for the expressible set of a sequence to have Lebesgue measure zero.

On summing to arbitrary real numbers

Following Erdős [1] we say a sequence {an}∞n=1 is irrational if the set { ∑ n≥1 1 ancn | cn ∈ N}, which we refer to henceforth as its expressible set, contains no rational numbers. In [1] it is shown that if limn→∞ a n n = ∞ and an ∈ N for all n ∈ N then ∑n≥1 a−1 n is an irrational number. From this Erdős deduces that the sequence {22n}∞n=1 is an irrational sequence. Thus its expressible set contains no rational numbers. In [2] it is shown that if an ∈ R+ for all n ∈ N and lim supn→∞ 1 n log2 log2 an < 1 then the expressible set of the sequence {an}∞n=1 contains an interval. It seems to be the case that in general finding the expressible set for the sequence {an}∞n=1 is not easy.