The Tu–Deng Conjecture holds Almost Surely
Lukas Spiegelhofer, Michael Wallner · The Electronic Journal of Combinatorics · 2019
The Tu–Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base $2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and$1\leqslant t<2^k-1$. Then\[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod 2^k-1, w(a)+w(b)