0
$\begingroup$

Let $$ f(x,N) = \sum_{0 \leq n\leq N} e(x n^2).$$

Weyl's inequality gives an estimate for $f(x,N)$ when $x$ is near a rational with small denominator. My question is:

What estimates are available for more general exponential sums of the form $$f(x,j,N) = \sum_{0 \leq n\leq N} n^j e(x n^2)? $$

Unlike the classical Weyl sum, the behavior of these functions do not appear to reduce to elementary Gauss sum estimates at rationals with small denominators. A related question is: what estimates are available for exponential sums of the form: $ \sum_{1\leq n \leq q} n^j e(n^2/q) $?

On the other hand, these sums do appear in the analytic number theory literature. For instance, see this paper of Hiary, although the interest there is in computational questions and not estimates.

Improve this question
$\endgroup$
4
  • 6
    $\begingroup$ Well, from summation by parts one can bound these expressions by $O(N^j)$ times the size of the largest unweighted partial sum up to $N$. I don't see any reason to expect significant improvements over this bound (though perhaps smoothing the sum can give some small gains). $\endgroup$
    – Terry Tao
    Commented Jun 22, 2016 at 2:31
  • 2
    $\begingroup$ That's what I needed, thanks! (When I was an undergrad, I recall Sarnak joking that a fair amount of his advising consisted of asking students if they had tried summation by parts...) $\endgroup$
    – Mark Lewko
    Commented Jun 22, 2016 at 3:19
  • $\begingroup$ Mate Wierdl and Joe Rosenblatt work quite hard on controlling exponential sums somewhat like this in their survey of subsequence ergodic theorems (published in a Cambridge Lecture Note volume, Ergodic Theory and its Connections with Harmonic Analysis). They use partial summation as Terry indicates. $\endgroup$
    – Anthony Quas
    Commented Jun 22, 2016 at 4:10
  • $\begingroup$ The basis of the Van-der-corput technique for estimating exponential sums relies upon relating a sum of the form $\sum g(n)e(f(n))$ to its "twisted continuous cousins" (done by effective truncation in the Poisson summation formula - see in Iwaniec and Kowalski's book). Estimating such continuous cousins is routine in harmonic analysis (stationary phase and such, see for example in Sogge's book). Nevertheless, the results are less optimal than Weyl's method (as the number theory involved is minimal), but asymptotically comparable. $\endgroup$
    – Asaf
    Commented Jun 22, 2016 at 10:10

0

Reset to default

You must log in to answer this question.