An $n H_k$-compressed searchable partial-sums data structure for static sequences of sublogarithmic positive integers

We consider the space needed to store a searchable partial-sums data structure with constant query time for a static sequence $S$ of $n$ positive integers in $o \left( \frac{\log n}{(\log \log n)^2} \right)$. Arroyuelo and Raman (2022) recently showed that such a structure can fit in $n H_0 (S) + o (n)$ bits. Starting with Ferragina and Venturini's (2007) $n H_k$-compressed representation of strings that supports fast random access, and augmenting it with sublinear data structures reminiscent of those Raman, Raman and Rao (2002) used in their succinct bitvectors, we slightly improve Arroyuelo and Raman's bound to $n H_k (S) + o (n)$ bits for $k \in o \left( \frac{\log n}{(\log \log n)^2} \right)$.