Title: The Tu--Deng Conjecture holds almost surely,
Abstract: 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 $1\leq t<2^k-1$. Then
\[\Bigl \lvert\Bigl\{(a,b)\in\bigl\{0,\ldots,2^k-2\bigr\}^2:a+b\equiv t\bmod
2^k-1, w(a)+w(b)<k\Bigr\}\Bigr \rvert\leq 2^{k-1}.\]
We prove that the Tu--Deng Conjecture holds almost surely in the following
sense: the proportion of $t\in[1,2^k-2]$ such that the above inequality holds
approaches $1$ as $k\rightarrow\infty$.
Moreover, we prove that the Tu--Deng Conjecture implies a conjecture due to
T.~W.~Cusick concerning the sum of digits of $n$ and $n+t$.