The digits of $n+t$

Abstract: We study the binary sum-of-digits function $s_2$ under addition of a constant $t$. For each integer $k$, we are interested in the asymptotic density $\delta(k,t)$ of integers $t$ such that $s_2(n+t)-s_2(n)=k$. In this talk, we consider the following two questions.

(1) Do we have \[ c_t=\delta(0,t)+\delta(1,t)+\cdots>1/2? \] This is a conjecture due to T. W. Cusick (2011).

(2) What does the probability distribution defined by $k\mapsto \delta(k,t)$ look like?

We prove that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M$ blocks of contiguous ones, where $M$ is effective. Our second theorem states that $\delta(j,t)$ usually behaves like a normal distribution, which extends a result by Emme and Hubert (2018).

This is joint work with Michael Wallner (TU Wien).

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

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