On a conjecture of Dekking : The sum of digits of even numbers

TL;DR

This paper proves Dekking's 1983 conjecture that the distribution of the sum of digits of even numbers in a given base is balanced over residue classes, showing no long-term bias exists.

Contribution

We establish the conjecture that the sum-of-digits function for even numbers is evenly distributed across residue classes, confirming the absence of a drift phenomenon.

Findings

The sum of digits of even numbers is evenly distributed modulo q.

The conjecture holds for all bases q ≥ 2.

No long-term bias in the distribution of sum-of-digits for even numbers.

Abstract

Let $q\geq 2$ and denote by $s_q$ the sum-of-digits function in base $q$. For $j=0,1,...,q-1$ consider $$# \{0 \le n < N : \;\;s_q(2n) \equiv j \pmod q \}.$$ In 1983, F. M. Dekking conjectured that this quantity is greater than $N/q$ and, respectively, less than $N/q$ for infinitely many $N$, thereby claiming an absence of a drift (or Newman) phenomenon. In this paper we prove his conjecture.