Modulo addition of independent random variables

Question

Let $X,Y,W$ be discrete random variables with support set $\{0,1,\ldots,M-1 \}$. Assume that $X$ and $Y$ are independent with respect to $W$ and $H(X)<H(Y)<\log M$ and $H(W)<\log M$.

Let $+$ be the modulo $M$ addition. Is it true that $H(X+W)<H(Y+W)$?

Answer 1

No. For example, let $M=8$, let $X$ be uniform over $(0,2,4,6)$, let $Y$ be uniform over $(0,1 ,2,3,4,5)$, let $M$ be uniform over $(0,1)$.

Then $H(Y)> H(X)$ but $H(X+M)>H(Y+M)$, because $X+M$ is uniform over the $M$ values but $Y+M$ is not.