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

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 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.