Approximate Discrete Entropy Monotonicity for Log-Concave Sums

It is proved that for any $n \geq 1,$ if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables then $$ h(X_1+\ldots+X_n + U_1+\ldots+U_n) = H(X_1+\ldots+X_n) + o(1), $$ as $H(X_1) \to \infty$, where $h$ stands for the differential entropy, $H$ dentoes the (discrete) Shannon entropy and $U_1,\ldots,U_n$ are independent continuous uniforms on $(0,1)$. As a corollary, it is shown that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: $$ H(X_1+\ldots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{1}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} - o(1) $$ as $H(X_1) \to \infty$. Explicit bounds for the $o(1)$-terms are provided.