Balka Richard: Null sequences with signs

ELTE Analízis tanszék szemináruma

Abstract: We answer a question of Mattila by proving that for any null sequence $(a_n)$ in $\mathbf{R}^d$ the sign sequences $(s_n)$ from the metric space $\{-1,1\}^{\mathbb N}$ for which $\sum s_n a_n$ converges form a set of maximal Hausdorff dimension. We show that we can differentiate between null sequences by using generalized Hausdorff measures. Understanding whether every vector arises as signed sums of a given null sequence is easy in dimension one, but remains widely open in higher dimensions.

This is joint work with Kornélia Héra, Gergely Kiss, and Benedict Sewell.