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On certain bilinear sums with modular square roots and applications

Stephan Baier

TL;DR

This work extends bounds on additive energies of modular square roots, deriving unconditional bounds for $E_2(r,j,M,H)$ and $E_4(r,j,M,H)$ via lattice-method techniques and algebraic transformations. These energy bounds feed into nontrivial estimates for bilinear sums with modular square roots, encapsulated in Theorems $\text{bilinearbound}$ and $\text{bilinearbound2}$, which leverage Weyl differencing and Poisson summation. A key application is a conditional improvement of the large sieve for square moduli at the critical point $N=Q^3$, asserted under a single energy-hypothesis on $E_4$. The results streamline previous hypotheses on additive energies and offer a roadmap for unconditional progress through polynomial-congruence analysis and exponential-sum methods, with potential broader applicability to bilinear forms involving arithmetic and analytic oscillations.

Abstract

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.

On certain bilinear sums with modular square roots and applications

TL;DR

This work extends bounds on additive energies of modular square roots, deriving unconditional bounds for and via lattice-method techniques and algebraic transformations. These energy bounds feed into nontrivial estimates for bilinear sums with modular square roots, encapsulated in Theorems and , which leverage Weyl differencing and Poisson summation. A key application is a conditional improvement of the large sieve for square moduli at the critical point , asserted under a single energy-hypothesis on . The results streamline previous hypotheses on additive energies and offer a roadmap for unconditional progress through polynomial-congruence analysis and exponential-sum methods, with potential broader applicability to bilinear forms involving arithmetic and analytic oscillations.

Abstract

We extend bounds on additive energies of modular square roots by Dunn, Kerr, Shparlinski, Shkredov and Zaharescu and apply these results to obtain bounds on certain bilinear exponential sums with modular square roots. From here, we make partial progress on the large sieve for square moduli.
Paper Structure (22 sections, 14 theorems, 179 equations)

This paper contains 22 sections, 14 theorems, 179 equations.

Table of Contents

  1. Introduction and main results
  2. Additive energies of modular square roots
  3. Preliminaries on lattices
  4. Bounding $E_2(r,j,M,H)$
  5. General strategy
  6. First moment of $I(d)$
  7. Algebraic transformations
  8. Contribution of $\tilde{r}=1$
  9. Partitioning of the $d$-sum
  10. The case $\lambda_1>1$
  11. The case $\lambda_1\leqslant 1$, $\lambda_2>1$
  12. The case $\lambda_2\leqslant 1$
  13. Concluding the proof
  14. Bounding $E_4(r,j,M,H)$
  15. Bounds on bilinear sums with modular square roots
  16. ...and 7 more sections

Key Result

Theorem 1

Let $Q,N\geqslant 1$, $M\in \mathbb{R}$ and $(a_n)_{M<n\leqslant M+N}$ be any sequence of complex numbers. Then where

Theorems & Definitions (21)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • Corollary 2
  • Theorem 6
  • Proposition 1: Minkowski
  • ...and 11 more