Signed null sequences and Hausdorff dimension

We investigate the convergence of signed null sequences of the form $$\sum _{n=1}^{\mathrm{\infty }}{\epsilon }_n{a}_n,{\epsilon }_n\in \{-1,1\},$$ where $(a_n)$ tends to zero in ${\mathbb{R}}^d$. Our main result shows that for any such sequence, the set of sign sequences yielding convergence has full Hausdorff dimension in the natural ultrametric topology. This answers a question of Mattila in the one-dimensional case, for which we provide an elementary proof. Moreover, if $(a_n)\mathrm{\notin }{\mathrm{\ell }}^1$ in one dimension, then for every $L\in \mathbb{R}$ the set of sign sequences with sum $L$ also has Hausdorff dimension $1$. In higher dimensions the analogous statement does not hold in full generality, but it is guaranteed if the sequence has $d$ linearly independent Lévy vectors.