The sum-of-digits function on arithmetic progressions

TL;DR

This paper investigates the behavior of the binary sum-of-digits function on arithmetic progressions, revealing that, aside from shifts, the set of m-tuples forming such sequences is highly complex and rich.

Contribution

It demonstrates that the sum-of-digits function in base 2 exhibits full complexity on arithmetic subsequences, extending understanding of its combinatorial properties.

Findings

The set of m-tuples appearing as arithmetic subsequences of s_2 has full complexity.

Up to a shift, the structure of these subsequences is highly intricate.

The results generalize previous knowledge about digit sum functions in arithmetic contexts.

Abstract

Let $s_2$ be the sum-of-digits function in base $2$, which returns the number of non-zero binary digits of a nonnegative integer $n$. We study $s_2$ alon g arithmetic subsequences and show that --- up to a shift --- the set of $m$-tuples of integers that appear as an arithmetic subsequence of $s_2$ has full complexity.