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Small Fractional Parts of Polynomials

About this Title

Wolfgang M. Schmidt

Publication: CBMS Regional Conference Series in Mathematics
Publication Year: 1977; Volume 32
ISBNs: 978-0-8218-3880-8 (print); 978-1-4704-2392-6 (online)
DOI: https://doi.org/10.1090/cbms/032

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Table of Contents

Front/Back Matter

Chapters

  • Heilbronn’s Theorem
  • The Heilbronn Alternative Lemma
  • Vinogradov’s Lemma
  • About Sums $\sum \| \xi _i \|^{-1}$
  • About Sums $\sum e(\alpha n^2)$
  • Proof of the Heilbronn Alternative Lemma
  • Fractional Parts of Polynomials
  • A General Alternative Lemma
  • Sums $\sum \| \xi _i \|^{-1}$ Again
  • Estimation of Weyl Sums
  • What Happens if the Weyl Sums are Large
  • Proof of the General Alternative Theorem
  • Simultaneous Approximation
  • A Reduction
  • A Vinogradov Lemma
  • Proof of the Alternative Lemma on Simultaneous Approximation
  • On max $\| \alpha _in^2 \|$
  • A Determinant Argument
  • Proof of the Three Alternatives Lemma
  • Quadratic Polynomials in Several Variables
  • Proofs for Quadratic Polynomials