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Squarefree numbers in short intervals

Mayank Pandey

Abstract

We show that there exists $η> 0$ such that the interval $[X, X + X^{\frac 15 - η}]$ contains a squarefree number for all large $X$. This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree number in $[X, X + cX^{\frac 15}\log X]$ for some $c > 0$ and all large $X$. We introduce a new technique to count lattice points near curves, which we use to bound in critical ranges the number of integers in a short interval divisible by a large square. This uses as an input Green and Tao's quantitative version of Leibman's theorem on the equidistribution of polynomial orbits in nilmanifolds.

Squarefree numbers in short intervals

Abstract

We show that there exists such that the interval contains a squarefree number for all large . This improves on an earlier result of Filaseta and Trifonov who showed that there is a squarefree number in for some and all large . We introduce a new technique to count lattice points near curves, which we use to bound in critical ranges the number of integers in a short interval divisible by a large square. This uses as an input Green and Tao's quantitative version of Leibman's theorem on the equidistribution of polynomial orbits in nilmanifolds.
Paper Structure (21 sections, 17 theorems, 156 equations)

This paper contains 21 sections, 17 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Setup and outline of proof
  4. Reduction to squarefree sieve bounds in short intervals and proof of the main theorem
  5. Proof of Proposition \ref{['prop:very_low_est']}
  6. Initial maneuvers for Proposition \ref{['prop:all_new_bounds']}
  7. Differencing identities
  8. An approximate parametrization of
  9. Outline of proof of Proposition \ref{['prop:all_new_bounds']}
  10. Final reductions
  11. Elementary estimates on points near curves
  12. Proof of bounds nontrivial for $\Delta = o(H^{\frac{1}{2}})$
  13. Some new differencings
  14. Obtaining nontrivial information about defects from linear patterns
  15. Small defects
  16. ...and 6 more sections

Key Result

Theorem 1.1

There exists $\eta > 0$ such that for $X^{\frac{1}{5} - \eta}\ll H\leqslant X$, we have that

Theorems & Definitions (28)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof : Proof of Theorem \ref{['thm:main']}
  • Theorem 2.3: FT2
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • proof : Proof of Proposition \ref{['prop:approx_param']}
  • Lemma 3.1
  • ...and 18 more