Discrete entropy of the integer part of a random variable Let $X$ be a real valued random variable. Of course, the integer part $\lfloor X \rfloor$ of $X$ is a discrete random variable taking values in $\mathbb{Z}$. We can therefore define its discrete entropy
\begin{equation}
H(\lfloor X \rfloor) = - \sum_{n\in\mathbb{Z}} \mathbb{P}( \lfloor X \rfloor = n ) \log( \mathbb{P}( \lfloor X \rfloor = n ) ), 
\end{equation}
which is in $[0,\infty]$ as a sum of nonnegative terms, since $- x \log x \geq 0$ for any $0 \leq x \leq 1$ (with the convention $0\log 0 = 0$).
I am looking for sufficient conditions such that $H(\lfloor X \rfloor ) < \infty$. For instance, is it sufficient to know that $X$ has a finite absolute moment $\mathbb{E}[|X|^p] < \infty$ for some $p>0$? Any condition of this type, possibly weaker, is welcome.
Motivation: There are strong connection between the differential entropy of $X$ (assuming $X$ has a pdf whose differential entropy is well-defined) and the discrete entropy of $\lfloor nX \rfloor$ when $n\rightarrow0$. This was the main topic of the 1959 paper from Alfred Rényi intitled On the dimension and entropy of probability distributions: I am questioning the assumptions under which the discrete entropy is well-defined.
 A: $\newcommand{\fx}{\lfloor X\rfloor}$ $\newcommand\Z{\mathbb{Z}}$
We shall prove more than requested: that $H(\fx)<\infty$ if $E\ln(1+|X|)<\infty$.
Indeed, let
$$p_n:=P(\fx=n),$$
so that
$$H(\fx)=-\sum_{n\in\Z}p_n\ln p_n.$$
Let $q\colon\mathbb R\to(0,\infty)$ be any function such that
$$\sum_{n\in\Z}q(n)=1\tag{1}$$
and
$$q(x)\le cq(\lfloor x\rfloor)\tag{2}$$
for some real $c>0$ and all real $x$.
Then by the Gibbs inequality for the Kullback–Leibler divergence between $(p_n)_{n\in\Z}$ and $(q(n))_{n\in\Z}$ we have
$$0\le KL((p_n)_{n\in\Z}||(q(n))_{n\in\Z})=\sum_{n\in\Z}p_n\ln\frac{p_n}{q(n)}=-H(\fx)+\sum_{n\in\Z}p_n\ln\frac1{q(n)},$$
whence, in view of (2),
$$H(\fx)\le\sum_{n\in\Z}p_n\ln\frac1{q(n)} \\
=\sum_{n\in\Z}\int_{[n,n+1)}P(X\in dx)\ln\frac1{q(n)} \\ 
\le\sum_{n\in\Z}\int_{[n,n+1)}P(X\in dx)\ln\frac c{q(x)} \\
=E\ln\frac c{q(X)}=\ln c+E\ln\frac1{q(X)}.$$
So,
$$H(\fx)<\infty\quad\text{if}\quad E\ln\frac1{q(X)}<\infty.$$
Taking here e.g. $q(x)=\frac C{(1+|x|)^2}$, where $C:=1/\sum_{n\in\Z}\frac1{(1+|x|)^2}[=\frac3{\pi ^2-3}]$, we have conditions (1) and (2) satisfied. So,
$$H(\lfloor X\rfloor)<\infty\quad\text{if}\quad E\ln(1+|X|)<\infty.$$
It follows that for any real $a>0$
$$H(\lfloor X\rfloor)<\infty\quad\text{if}\quad E|X|^a<\infty,$$
as was initially desired.
A: Since $\lfloor X\rfloor$ has finite entropy if and only if $|\lfloor X\rfloor|$ has finite entropy, it suffices to consider random variables taking values in the natural numbers. Write $p_n$ for $\mathbb P(X=n)$ (so that $\sum_n p_n=1$). We have $X\in L^q$ if and only if $\sum p_n n^q<\infty$.
Suppose $X\in L^q$ so that $\sum p_n n^q<\infty$.
Then let $S_1=\{n\colon p_n<\frac{1}{n^2}\}$ and $S_2=\{n\colon p_n\ge \frac 1{n^2}\}$.
We have
$$
H(X)=\sum_n -p_n\log p_n=-\sum_{n\in S_1}p_n\log p_n-\sum_{n\in S_2}p_n\log p_n.
$$
Since $-t\log t$ is increasing on $[0,\frac 1e]$, the first sum is bounded above by
$$
\sum_{n\in S_1}\frac{2\log n}{n^2}<\infty.
$$
There exists an $n_0$ so that for $n\ge n_0$, $2\log n<n^q$. For $n\in S_2$ such that $n\ge n_0$, $-\log p_n<2\log n<n^q$, so that
$$
-\sum_{n\in S_2,\,n\ge n_0}p_n\log p_n\le 
\sum_{n\in S_2,\,n\ge n_0}p_n n^q<\infty.
$$
Hence $H(X)<\infty$. (This trick appears in a couple of papers of mine: one with Ciprian Demeter in NYJM and another more recent preprint with Tamara Kucherenko and Christian Wolf).
A: Using (say) decimal notation, ASCII encoding, and a delimiter symbol such as a space or comma, as well as the law of large numbers, one can almost surely encode $N$ independent copies of $\lfloor X \rfloor$ using $O( N {\bf E} \log( 2 + |X| ) ) + o(N)$ bits.  Applying the Shannon source coding theorem, we conclude that
$$ {\bf H}( \lfloor X \rfloor ) \ll {\bf E} \log(2 + |X| )$$
which by Jensen's inequality also gives
$$ {\bf H}( \lfloor X \rfloor ) \ll_p \log(2 + {\bf E} |X|^p)$$
for any $0 < p < \infty$.
