[Paper Review] New estimates of double trigonometric sums with exponential functions

M. Z. Garaev, A. A. Karatsuba|ArXiv.org|Apr 1, 2005

Analytic Number Theory Research4 references4 citations

TL;DR

This paper presents a new bound for double exponential sums involving multiplicative characters and exponential functions over finite fields, using a refined application of the Vinogradov mean value method. The key contribution is a significant improvement in the exponent of $ p $, achieving nontrivial estimates when $ |\/mathcal{X}| \geq |\mathcal{Y}| \geq p^{3/4+\varepsilon} $, surpassing prior results in the literature for such sums with general coefficient sequences.

ABSTRACT

We establish a new bound for the exponential sum \begin{eqnarray*} \sum_{x\in\mathcal{X}}\Big|\sum_{y\in \mathcal{Y}}γ(y)\exp(2πi a λ^{xy}/p)\Big|, \end{eqnarray*} where $λ$ is an element of the residue ring modulo a large prime number $p,$ $\mathcal{X}$ and $\mathcal{Y}$ are arbitrary subsets of the residue ring modulo $p-1$ and $γ(n)$ are any complex numbers with $|γ(n)| \le 1.$ In particular, we improve several previously known bounds.

Motivation & Objective

• To improve existing bounds for double exponential sums of the form $ \sum_{x\in\mathcal{X}} \left| \sum_{y\in\mathcal{Y}} \gamma(y) \exp(2\pi i a \lambda^{xy}/p) \right| $, where $ \lambda $ has order $ T $ modulo a prime $ p $.
• To extend and refine the method of [10] by introducing a new ingredient in the estimation process, enabling stronger bounds for sums over arbitrary subsets $ \mathcal{X}, \mathcal{Y} \subset \mathbb{Z}_{p-1} $.
• To provide improved estimates that are nontrivial even for sparse sets $ \mathcal{Y} $, particularly when $ |\mathcal{Y}| \geq p^{3/4+\varepsilon} $, surpassing previous thresholds such as $ p^{7/8+\varepsilon} $.
• To establish a general bound that applies to both multiplicative character sums and exponential sums over elliptic curves, as suggested by applications in [3] and [10].

Proposed method

• The authors use a refined version of the Vinogradov mean value method, incorporating a new decomposition technique based on divisor sums and level sets of multiplicative characters.
• They define a set $ \mathcal{L}_d $ of indices $ y $ such that $ dy \in \mathcal{Y} $ and $ (dy, p-1) = d $, enabling a dyadic decomposition over divisors $ d \mid p-1 $.
• For small $ d < Tp^{-1/2}(\log p)^{-4k} $, they apply a new hybrid estimate using $ \ell $-level sets and Cauchy-Schwarz to control the sum over $ x $ and $ y $, leveraging cancellation in exponential sums.
• The method involves bounding the number of solutions to a system of congruences $ uv_j \equiv t \pmod{p} $, using $ \delta(uv) $ to count solutions and applying $ L^2 $-type estimates via the large sieve or exponential sum bounds.
• A key innovation is the use of a parameter $ k $ to interpolate between different bounds, allowing the exponent of $ p $ to be optimized asymptotically as $ k \to \infty $.
• The final bound is obtained by summing over all divisors $ d \mid p-1 $, separating the contribution from small and large $ d $, and using the $ \tau(p-1) = p^{o(1)} $ bound on the number of divisors.

Experimental results

Research questions

• RQ1 Can the bound for double exponential sums with arbitrary complex coefficients $ \gamma(y) $, $ |\gamma(y)| \leq 1 $, be improved beyond the state-of-the-art $ p^{7/8+o(1)} $ exponent?
• RQ2 What is the optimal exponent of $ p $ in the bound for $ W_a(\gamma; T; \mathcal{X}, \mathcal{Y}) $ when $ \mathcal{X}, \mathcal{Y} $ are arbitrary subsets of $ \mathbb{Z}_{p-1} $, and how does it depend on $ k $?
• RQ3 Can the new method be extended to bound character sums over elliptic curves or other algebraic varieties, as suggested by [3]?
• RQ4 What is the threshold for $ |\mathcal{Y}| $ below which the new bound remains nontrivial, and how does it compare to previous results?

Key findings

• The paper establishes a new bound: $ W_a(\gamma; T; \mathcal{X}, \mathcal{Y}) \ll \frac{|\mathcal{X}|^{1-1/(2k)}|\mathcal{Y}|^{1-1/(2k+2)} p^{1/(2k) + 3/(4k+4) + o(1)}}{T^{1/(2k+2)}} $, valid for any fixed $ k \geq 1 $.
• For $ T = p-1 $, the bound becomes $ W_a(\gamma; p-1; \mathcal{X}, \mathcal{Y}) \ll |\mathcal{X}|^{1-1/(2k)}|\mathcal{Y}|^{1-1/(2k+2)} p^{1/(2k) + 1/(4k+4) + o(1)} $, improving the previous $ p^{7/8+o(1)} $ exponent.
• When $ k \to \infty $, the bound becomes nontrivial for $ |\mathcal{X}| \geq |\mathcal{Y}| \geq p^{3/4+\varepsilon} $, improving on the prior threshold of $ p^{7/8+\varepsilon} $.
• For the case $ T \ll p $, the bound $ W_a(\gamma; T; \mathcal{X}_T, \mathcal{Y}_T) \ll |\mathcal{X}_T|^{1-1/(2k)}|\mathcal{Y}_T|^{1-1/(2k+2)} T^{1/(2k)} p^{1/(4k+4)+o(1)} $ improves on the previous $ p^{1/8+o(1)} $ exponent for $ T $-torsion sums.
• The method yields a new bound $ W_a(\gamma; T; \mathcal{X}, \mathcal{Y}) \ll \frac{|\mathcal{X}|^{1/2}|\mathcal{Y}|^{3/4} p^{7/8+o(1)}}{T^{1/4}} $, which improves on earlier results for thin sets $ \mathcal{Y} $ with $ |\mathcal{Y}| \leq p^{1-c} $.