Dense signed sums of non-integer powers

Abstract

We prove that if $j>0$ is not an integer, then there is a choice of signs ${\epsilon }_k\in \{\pm 1\}$ such that the partial sums ${\sum }_{k=1}^N{\epsilon }_k{k}^j$ are dense in $\mathbb{R}$. The proof groups consecutive terms into Thue--Morse blocks, whose Prouhet--Tarry--Escott cancellation produces nonzero block sums tending to zero but with divergent total variation. A standard steering argument then chooses block signs so that the resulting partial sums visit arbitrarily small neighbourhoods of every real number.