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.

翻译：摘要：本文证明陶哲轩（2010）提出的猜想对整数集上的对数凹随机变量成立：对于任意 $n \geq 1$，若 $X_1,\ldots,X_n$ 为独立同分布的整数值对数凹随机变量，则当 $H(X_1) \to \infty$ 时，有 $$ H(X_1+\cdots+X_{n+1}) \geq H(X_1+\cdots+X_{n}) + \frac{1}{2}\log{\Bigl(\frac{n+1}{n}\Bigr)} - o(1)，$$ 其中 $H$ 表示（离散）香农熵。通过证明若 $U_1,\ldots,U_n$ 为 $(0,1)$ 上独立的连续均匀分布，则当 $H(X_1) \to \infty$ 时，有 $$ h(X_1+\cdots+X_n + U_1+\cdots+U_n) = H(X_1+\cdots+X_n) + o(1)，$$ 其中 $h$ 表示微分熵，从而将该问题转化为连续情形。本文还给出了 $o(1)$ 项的显式界。