  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.
