The binary digits of n plus t

The binary sum-of-digits function s counts the number of ones in the binary expansion of a nonnegative integer. For any nonnegative integer t, T. W. Cusick defined the asymptotic density c(t) of integers n >= 0 such that s(n + t) >= s(n) In 2011, he conjectured that c(t) > 1/2 for all t - the binary sum of digits should, more often than not, weakly increase when a constant is added. In this paper, 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 maximal blocks of contiguous ones, leaving open only the "initial cases" - few maximal blocks of ones - of this conjecture. Moreover, we sharpen a result by Emme and Hubert (2019), 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 maximal blocks of ones in the binary expansion of t grows.