Block occurrences in the binary expansion of n and n+t

Abstract: Let $s(n)$ denote the sum of binary digits of a nonnegative integer $n$. In 2012 Cusick asked whether for every nonnegative integer $t$ the set of $n$ satisfying $s(n+t) \geq s(n)$ has natural density strictly greater than $1/2$. So far it is known that the answer is affirmative for almost all $t$ (in the sense of density) and $s(n+t) - s(n)$ has approximately Gaussian distribution. During the talk we consider an analogue of this problem concerning the function $r(n)$, which counts the occurrences of the block $11$ in the binary expansion of $n$. In particular, we prove that the distribution of $r(n+t)-r(n)$ is approximately Gaussian as well. We also discuss a generalization to an arbitrary block of binary digits. This is a joint work with Lukas Spiegelhofer.

dynamical systemsnumber theory

Audience: researchers in the topic

( paper | slides )

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One World Numeration seminar

Series comments: Description: Online seminar on numeration systems and related topics

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