SUM-PRODUCT PHENOMENA: p-ADIC CASE.

NEW

Problem: Reconstructed statement: for every finite field $\mathbb F_q$, for independent $\mathbb F_q$-valued random variables $X,Y$ and $\alpha$, with $\alpha$ uniform on $\mathbb F_q^\times$, and $Z_{\mathbb F_q}$ uniform on $\mathbb F_q$, is there a universal constant $c>0$ such that

$$H(X+\alpha Y\mid \alpha)\ge \min\{H(X)+H(Y),H(Z_{\mathbb F_q})\}-c?$$

The text does not explicitly state independence; without independence it is trivially false. The counterexample below refutes even this natural independent version.

Result: No such universal constant exists.

Use natural-log entropy; any other fixed base only rescales constants. Let $U,V$ be independent uniform random variables on $\mathbb F_q$, let $B,C$ be independent Bernoulli$(1/2)$, and let $\alpha$ be independent uniform on $\mathbb F_q^\times$. Define

$$X=BU,\qquad Y=CV.$$

Thus $X,Y,\alpha$ are independent.

For $0<r<1$, let $\mu_{r,q}=r\delta_0+(1-r)\mathrm{Unif}(\mathbb F_q)$. Then

$$H(\mu_{r,q})=(1-r)\log q+h(r)+o(1),$$

where $h(r)=-r\log r-(1-r)\log(1-r)$.

Here $X,Y\sim \mu_{1/2,q}$, so

$$H(X)+H(Y)=\log q+2\log 2+o(1)>\log q$$

for all sufficiently large $q$. Since $Z_{\mathbb F_q}$ is uniform,

$$H(Z_{\mathbb F_q})=\log q,$$

so the right-hand side becomes $\log q-c$.

Now fix $\alpha=a\neq 0$. If $B=C=0$, which has probability $1/4$, then

$$X+aY=0.$$

In all other three cases, $X+aY$ is uniform on $\mathbb F_q$: it is $U$, or $aV$, or $U+aV$. Hence

$$X+\alpha Y\mid \alpha=a \sim \mu_{1/4,q}.$$

Therefore

$$H(X+\alpha Y\mid \alpha)=H(\mu_{1/4,q}) =\frac34\log q+h(1/4)+o(1).$$

Thus

$$\log q-H(X+\alpha Y\mid \alpha) =\frac14\log q-h(1/4)+o(1)\to\infty.$$

Given any fixed $c$, choose $q$ large enough that this difference exceeds $c$. Then

$$H(X+\alpha Y\mid \alpha)<\log q-c =\min\{H(X)+H(Y),H(Z_{\mathbb F_q})\}-c,$$

contradicting the proposed inequality.

So Question 8, as stated for arbitrary random variables, is false. The obstruction is not dependence or a boundary case: $X,Y,\alpha$ are independent and nonconstant. The failure comes from non-flat distributions with large atoms.

Citation: No prior disproof is used here. Source question: Alireza Salehi Golsefidy, “Sum-product phenomena: p-adic case,” arXiv:1602.00400, Question 8.