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For a nonnegative integer $t$, let $c_t$ be the asymptotic density of naturalnumbers $n$ for which $s(n + t) \geq s(n)$, where $s(n)$ denotes the sum ofdigits of $n$ in base $2$. We prove that $c_t > 1/2$ for $t$ in a set ofasymptotic 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 hasseveral 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 ofthe main result is based on Chebyshev's inequality and the asymptotic analysisof a trivariate rational function, using methods from analytic combinatorics.

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