The digits of n+t

The binary sum-of-digits function $s(n)$ counts the number of $\mathtt 1$s in the binary expansion of a nonnegative integer $n$. T.~W.~Cusick defined the asymptotic density \[ c_t=\lim_{N\rightarrow \infty} \frac 1N \lvert\{0\leq n1/2$ for all $t$. We prove that there exists an explicit constant $M_0$ such that indeed $c_t>1/2$ if the binary expansion of $t$ contains at least $M_0$ blocks of contiguous $\mathtt 1$s. Moreover, we sharpen a result by Emme and Hubert (2018), proving that the difference $s(n+t)-s(n)$ behaves according to a Gaussian distribution, up to an error tending to $0$ as the number of blocks in the binary expansion of $t$ grows.