Approximate Discrete Entropy Monotonicity for Log-Concave Sums

It is proven that a conjecture of Tao (2010) holds true for log-concave random variables on the integers: For every $n \geq 1$, if $X_1,\ldots,X_n$ are i.i.d. integer-valued, log-concave random variables, then $$H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{1}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} - o(1)$$ as $H(X_1) \to \infty$, where $H$ denotes the (discrete) Shannon entropy. The problem is reduced to the continuous setting by showing that if $U_1,\ldots,U_n$ are independent continuous uniforms on $(0,1)$, then $$h(X_1+\cdots+X_n + U_1+\cdots+U_n) = H(X_1+\cdots+X_n) + o(1)$$ as $H(X_1) \to \infty$, where $h$ stands for the differential entropy. Explicit bounds for the $o(1)$-terms are provided.