Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sieving with square conditions and applications to Hilbert cubes in arithmetic sets

Rainer Dietmann, Christian Elsholtz, Imre Ruzsa

Abstract

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo $p^2$ to more suitable residue information modulo $p$, so that we can successfully apply the sieve. 2) The applications to Hilbert cubes are of interest in their own right: We study the maximal dimension of Hilbert cubes in various multiplicatively defined sets. For the squareful numbers in $[1,N]$ we achieve an upper bound of the dimension of $d=O(\log N)$. The same upper bounds follow for multiplicative semigroups of integers defined by a positive proportion of the primes, and the set of integers representable by an irreducible positive definite binary quadratic form. Eventually, making use of the sun flower lemma we give an improvement on the maximal dimension $d$ of subset sums in the set of pure powers in $[1,N]$.

Sieving with square conditions and applications to Hilbert cubes in arithmetic sets

Abstract

The purpose of this paper is twofold: 1) Applications of Gallagher's larger sieve modulo prime squares do not work. In some relevant cases we can transform the residue class information modulo to more suitable residue information modulo , so that we can successfully apply the sieve. 2) The applications to Hilbert cubes are of interest in their own right: We study the maximal dimension of Hilbert cubes in various multiplicatively defined sets. For the squareful numbers in we achieve an upper bound of the dimension of . The same upper bounds follow for multiplicative semigroups of integers defined by a positive proportion of the primes, and the set of integers representable by an irreducible positive definite binary quadratic form. Eventually, making use of the sun flower lemma we give an improvement on the maximal dimension of subset sums in the set of pure powers in .
Paper Structure (11 sections, 14 theorems, 29 equations)

This paper contains 11 sections, 14 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Aims
  3. Notation
  4. Old and new tools from additive combinatorics
  5. Main results: application to problems on Hilbert cubes in multiplicatively defined sets
  6. Gallagher's larger sieve modulo powers
  7. Proofs
  8. Proof of Theorem \ref{['Ruzsa']} and Corollary \ref{['schwarzwald']}
  9. Application of the larger sieve
  10. Subset sums in pure powers, proof of Theorem \ref{['thm:pure-powers']}
  11. Proofs of Corollary \ref{['cor:sumsof2squares']} and Theorem \ref{['2nddensitytheorem']}

Key Result

Lemma 1.1

Let $p$ be a prime, ${\mathcal{B}} \subset {\mathbb{Z}}/p{\mathbb{Z}}$ with $|{\mathcal{B}}|>2\sqrt{p}$ and $a \in {\mathbb{Z}}/p{\mathbb{Z}}$. Then there exists a non-empty ${\mathcal{A}} \subset {\mathcal{B}}$ such that $\sum({\mathcal{A}}) \equiv a \pmod p$.

Theorems & Definitions (23)

  • Lemma 1.1: Olson Olson:1968, Theorem 1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4: Hilbert cubes in semigroups, motivated by norm forms
  • Remark
  • Corollary 1.5: Hilbert cubes in squareful numbers
  • Theorem 1.6: Subsetsums in pure powers
  • Corollary 1.7: Hilbert cubes in binary quadratic forms
  • Theorem 1.8: Hilbert cubes in multiplicative semigroups
  • Lemma 1.9: See Gallagher:1971
  • ...and 13 more