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Let $p$ be a large prime. I would like to say that the multi-set $[1,p^{1-\varepsilon}]^2 = \{ab \mod p: a, b \in [1,p^{1-\varepsilon}]\}$ is close to uniformly distributed, i.e. that every nonzero residue class mod $p$ occurs with almost equal frequency.

Quantitatively, is it true that (almost?) every nonzero residue class appears at most $O(p^{1-2\varepsilon})$ times?

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    $\begingroup$ True for each $\epsilon < 1/4$ by the bound on Kloosterman sums. Probably true for each $\epsilon < 1/2$ but I don't think we know how to prove this. $\endgroup$ Commented Sep 7, 2015 at 15:58

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Here is a more explicit version of Noam Elkies's comment.

Theorem. Let $p>2$ be a prime number. Let $\mathcal{U},\mathcal{V}\subseteq\{1,2,\dots,p-1\}$ be two intervals, and let $r\in\{1,2,\dots,p-1\}$ be a nonzero residue modulo $p$. Then $$\Biggl|\sum_{\substack{u\in\mathcal{U},\ v\in\mathcal{V} \\ uv\equiv r \pmod{p}}} 1-\frac{|\mathcal{U}||\mathcal{V}|}{p-1}\Biggr|<2p^{1/2}(\log p)^2.$$

You can find a proof from scratch here (see Theorem 9).

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If you're OK with "almost every", we can use a $L^2$ method and multiplicative character sum bounds to do better.

Let's estimate $$ \sum_{c \in \mathbb F_p^\times} \left( \left[\sum_{a,b \in I | ab=c } 1\right] - p^{1-2\epsilon}\right)^2 $$

by Plancherel $$= \frac{1}{p-1} \sum_{\chi \in \left( \mathbb F_p^{\times} \right)^{\vee}, \chi \neq 1} \left | \sum_{a,b \in I} \chi(ab) \right|^2 $$

and multiplicativity$$= \frac{1}{p-1} \sum_{\chi \in \left( \mathbb F_p^{\times} \right)^{\vee}, \chi \neq 1} \left | \sum_{a\in I} \chi(a) \right|^4 $$

Now assume we have a nontrivial character sum bound $\left| \sum_{a \in I} \chi(a) \right| \leq N$ for all nontrivial characters $\chi$. Then we continue:

$$ \leq \frac{1}{p-1} \sum_{\chi \in \left( \mathbb F_p^{\times} \right)^{\vee}, \chi \neq 1} N^2 \left | \sum_{a\in I} \chi(a) \right|^2 \leq \frac{1}{p-1} \sum_{\chi \in \left( \mathbb F_p^{\times} \right)^{\vee} } N^2 \left | \sum_{a\in I} \chi(a) \right|^2 = N^2 p^{1-\epsilon} $$

Then we see that

$$\left[\sum_{a,b \in I | ab=c } 1\right] \leq 2 p^{1-2\epsilon} $$

for all but

$$\frac{ N^2 p^{1-\epsilon}}{ p^{2-4 \epsilon} } = N^2 p^{3\epsilon-1}$$

This is $\ll p$ as long as $N \ll p^{1 - 3\epsilon/2}$. Using the Pólya-Vinagradov bound where $N \approx p^{1/2}$, this works for $\epsilon <1/3$. Using the Burgess bound we can do slightly better, but not get all the way to $\epsilon = 1/2$, which would require $N= p^{1/4}$ - full square root cancellation, outside of reach.

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  • $\begingroup$ Should it be $1/p$ in front of Plancherel instead of $1/(p-1)$? $\endgroup$
    – Xiaoyu He
    Commented Sep 9, 2015 at 1:58
  • $\begingroup$ @alkjash I don't think so, because I'm doing it on a group of size $p-1$, the multiplicative group. $\endgroup$
    – Will Sawin
    Commented Sep 9, 2015 at 2:01
  • $\begingroup$ Yea, sorry. I just realized that the true mean should be $|I|^2/(p-1)$, not $|I|^2/p$. $\endgroup$
    – Xiaoyu He
    Commented Sep 9, 2015 at 2:03

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