4
$\begingroup$

Let $p$ be a prime. Let $\omega$ be a $ p $-th root of unity. We know that $ \chi_\alpha (x) = \omega^{\alpha\cdot x} $ are the additive characters of $ \mathbb{Z}_p $.

I have a question about bounding incomplete sums of these characters.

Let $ T $ be a subset of $ \mathbb{Z}_p $ of size $ (p-1)/2 $. I want statistics on sums of the form: $$ \sum_{t \in T} \chi_\alpha(t) $$

In the worst-case, we know that this sum can be as large as $ c\cdot p $ where $ c $ is some constant. (e.g. when $ \alpha = 0 $ or $ \alpha =1 $ and $ T = \{ 1, 2, \dots, (p-1)/2 \}$ )

Can we say something stronger about the expectation? That is, for every subset $ T $, the expectation: $$ \mathbb{E}_{\alpha \gets \mathbb{Z}_p} \sum_{t \in T} \chi_\alpha(t) $$

For example, is it bounded by $ p^{\frac{1}{2}} $?

$\endgroup$

1 Answer 1

2
$\begingroup$

The trivial estimate here would be $$ \mathbb E\,\Bigg|\sum_{t\in T}\chi_\alpha(t)\Bigg| \le \Bigg( \mathbb E\,\Bigg| \sum_{t\in T} \chi_\alpha(t) \Bigg|^2 \Bigg )^{1/2} = \Bigg( \sum_{t_1,t_2\in T} \frac1p\sum_{\alpha\in\mathbb Z_p} \omega^{\alpha\cdot(t_1-t_2)} \Bigg)^{1/2} = \sqrt{|T|} \le p^{1/2}. $$ Does this answer your question?

$\endgroup$
2
  • 3
    $\begingroup$ One can see that this bound is sharp to within a constant factor, as long as $|T| \leq p/2$, by considering a random subset $T$. $\endgroup$
    – Will Sawin
    Commented Oct 14, 2017 at 11:21
  • 1
    $\begingroup$ Also, without the absolute values, one has ${\mathbb E} \sum_{t \in T} \chi_\alpha(t) = 1_{0 \in T}$. $\endgroup$
    – Terry Tao
    Commented Oct 14, 2017 at 16:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.