SUM-PRODUCT PHENOMENA: p-ADIC CASE.

NEW

Problem: The reconstructed question is whether Theorem 1 remains true with constants independent of the finite extension $K/\mathbb Q_p$: for each small $\varepsilon>0$, do there exist $\delta>0$, $C\in\mathbb N$, and $N_0$, depending only on $\varepsilon$, such that the conclusion of Theorem 1 holds for every finite extension $K/\mathbb Q_p$, every $N\ge N_0$, and every $A\subseteq\mathcal O_K$ with

$$|\pi_{\varpi^N}(A)|\ge |\mathcal O_K/\varpi\mathcal O_K|^{N\varepsilon}?$$

Here the conclusion includes the existence of a subfield $K_0\subseteq K$, $a\in\mathcal O_K$, and integers $N_1,N_2$ such that

$$N_1+\lfloor \delta N\rfloor\le N_2\le CN,\qquad \pi_{\varpi^{N_2}}(\mathcal O_{K_0}a)\subseteq \pi_{\varpi^{N_2}}(\langle A\rangle_C),$$

and

$$|\mathcal O_{K_0}/p\mathcal O_{K_0}| \ge |\mathcal O_K/p\mathcal O_K|^{\varepsilon/4}.$$

Result: This unrestricted statement is false.

Fix $0<\varepsilon\le 1/2$, and put $r=\lfloor 1/\varepsilon\rfloor$, so $r\ge2$ and $1/r\ge\varepsilon$. Suppose uniform constants $\delta,C,N_0$ existed. Choose $N\ge N_0$ with $\lfloor\delta N\rfloor\ge2$, and then choose a prime $n>\max(CN,4/\varepsilon)$. Let $p=2$, and let

$$K=\mathbb Q_2(\varpi),\qquad \varpi^n=2.$$

The polynomial $x^n-2$ is Eisenstein, so $[K:\mathbb Q_2]=n$, $\varpi$ is a uniformizer, and the residue field is $\mathbb F_2$. Since $n$ is prime, the only subfields of $K$ are $\mathbb Q_2$ and $K$.

Let

$$A=\mathbb Z_2[\varpi^r]\subseteq\mathcal O_K.$$

Because $N<n$, modulo $\varpi^N$ the image of $A$ has $\mathbb F_2$-basis

$$1,\varpi^r,\varpi^{2r},\dots,\varpi^{\lfloor (N-1)/r\rfloor r}.$$

Thus

$$|\pi_{\varpi^N}(A)|=2^{\lfloor (N-1)/r\rfloor+1} =2^{\lceil N/r\rceil} \ge 2^{N\varepsilon} =|\mathcal O_K/\varpi\mathcal O_K|^{N\varepsilon}.$$

So $A$ satisfies the hypothesis.

Since $A$ is a subring, $\langle A\rangle_C\subseteq A$. If the conclusion held, the box-dimension condition on $K_0$ would force $K_0=K$: indeed,

$$|\mathcal O_K/2\mathcal O_K|=2^n,$$

whereas for $K_0=\mathbb Q_2$ one has $|\mathbb Z_2/2\mathbb Z_2|=2<2^{n\varepsilon/4}$. Hence $\mathcal O_{K_0}a=\mathcal O_Ka=\varpi^{N_1}\mathcal O_K$.

Now $N_2\le CN<n$. Modulo $\varpi^{N_2}$, every nonzero element of $A=\mathbb Z_2[\varpi^r]$ has leading valuation divisible by $r$. But

$$N_2-N_1\ge \lfloor\delta N\rfloor\ge2,$$

so the interval $[N_1,N_2)$ contains some integer $t$ not divisible by $r$. Then $\varpi^t\in\varpi^{N_1}\mathcal O_K$, but its class modulo $\varpi^{N_2}$ cannot lie in $\pi_{\varpi^{N_2}}(A)$. This contradicts

$$\pi_{\varpi^{N_2}}(\varpi^{N_1}\mathcal O_K) \subseteq \pi_{\varpi^{N_2}}(\langle A\rangle_C).$$

Therefore Theorem 1 cannot hold with no restriction on $K$. Some information controlling the ramification degree/degree of $K$ is genuinely necessary.

Citation: The reconstructed statement is from Salehi Golsefidy, “Sum-product phenomena: $p$-adic case,” arXiv:1602.00400, Question 5. The counterexample above is elementary and no existing resolution is being invoked.