On a Conjecture of Cusick Concerning the Sum of Digits of n and n + t

TL;DR

This paper proves that for most nonnegative integers t, the proportion of numbers n with the sum of binary digits of n+t at least as large as that of n exceeds one-half, partially confirming Cusick's conjecture.

Contribution

It establishes that the density c_t exceeds 1/2 for a set of t with asymptotic density 1, advancing understanding of Cusick's conjecture.

Findings

c_t > 1/2 for a set of t with asymptotic density 1

Partial confirmation of Cusick's conjecture for most t

Uses analytic combinatorics and rational function analysis

Abstract

For a nonnegative integer $t$, let $c_t$ be the asymptotic density of natural numbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum of digits of $n$ in base $2$. We prove that $c_t > 1/2$ for $t$ in a set of asymptotic density $1$, thus giving a partial solution to a conjecture of T. W. Cusick stating that $c_t > 1/2$ for all t. Interestingly, this problem has several equivalent formulations, for example that the polynomial $X(X + 1)\cdots(X + t - 1)$ has less than $2^t$ zeros modulo $2^{t+1}$. The proof of the main result is based on Chebyshev's inequality and the asymptotic analysis of a trivariate rational function, using methods from analytic combinatorics.