From Cusick's conjecture to deletion rigidity in binary digit problems

Abstract: Let $s_2(n)$ denote the sum of the binary digits of $n$. Cusick's conjecture asserts that for every positive integer $t$, $\mathrm{dens}\{n \ge 0: s_2(n+t) \ge s_2(n)\} > 1/2$. In this talk, I will present a proof based on a first-exit interpretation of binary addition, in which the relevant distributions are realized through stopped random walks on subsequence ideals of binary words. I will then briefly discuss extensions of this viewpoint to the Tu–Deng conjecture, cyclic deletion rigidity and Macaulay shadows, as well as sharp extremal asymptotics for Cusick's bias at fixed Hamming weight.

dynamical systemsnumber theory

Audience: researchers in the topic

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One World Numeration seminar

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