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The large sieve for square moduli, revisited

Stephan Baier

TL;DR

This work revisits the large sieve for square moduli and demonstrates conditional improvements based on additive-energy hypotheses for modular square roots. By combining Wolke’s sparse-moduli strategy, Poisson summation in a targeted variable, Weyl differencing, and novel energy bounds E2,E4 (and their differences), the authors obtain a nontrivial exponent saving in the main sum when N=Q^3 and under hypotheses H2 and H3. A key outcome is a bound of the form $\sum_{q\le Q}\sum_{(a,q)=1}^{q^2} |\sum_{M<n\le M+N} a_n e(na/q^2)|^2 \ll Q^{7/2-1/135} Z$, representing a reduction by a factor $Q^{1/135}$ over prior conditional bounds, with further refinements in special parameter regimes. The work also highlights intriguing connections to short character sums via Poisson completion and outlines concrete avenues to potentially remove the conditionality by establishing unconditional energy bounds and broader applicability to general moduli.

Abstract

We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots.

The large sieve for square moduli, revisited

TL;DR

This work revisits the large sieve for square moduli and demonstrates conditional improvements based on additive-energy hypotheses for modular square roots. By combining Wolke’s sparse-moduli strategy, Poisson summation in a targeted variable, Weyl differencing, and novel energy bounds E2,E4 (and their differences), the authors obtain a nontrivial exponent saving in the main sum when N=Q^3 and under hypotheses H2 and H3. A key outcome is a bound of the form , representing a reduction by a factor over prior conditional bounds, with further refinements in special parameter regimes. The work also highlights intriguing connections to short character sums via Poisson completion and outlines concrete avenues to potentially remove the conditionality by establishing unconditional energy bounds and broader applicability to general moduli.

Abstract

We revisit the large sieve for square moduli and obtain conditional improvements under hypotheses on higher additive energies of modular square roots.

Paper Structure

This paper contains 36 sections, 22 theorems, 336 equations.

Table of Contents

  1. Current state of the art
  2. Review of the classical large sieve
  3. Known results on the large sieve with sparse sets of moduli
  4. Known results on the large sieve for square moduli
  5. Applications
  6. Additive energies of modular square roots and main result
  7. Preliminaries from the theory of exponential sums
  8. Wolke's approach
  9. General approach
  10. Wolke's treatment of prime moduli
  11. Application of Wolke's approach to square moduli
  12. Previous ways to estimate $P(x)$
  13. Direct count
  14. Poisson summation in $m$ and $q$
  15. Advantages and disadvantages
  16. ...and 21 more sections

Key Result

Theorem 1

Let $Q,N\in \mathbb{N}$, $M\in \mathbb{Z}$ and $(a_n)_{M<n\le M+N}$ be any sequence of complex numbers. Then

Theorems & Definitions (32)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Theorem 1.1. in KSSZ
  • Theorem 4: Theorem 1.2. in KSSZ
  • Theorem 8
  • Proposition 9: Weyl differencing
  • proof
  • Proposition 10: Poisson summation
  • proof
  • Proposition 11
  • ...and 22 more