# Small Fractional Parts of Polynomials

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*Wolfgang M. Schmidt*

A co-publication of the AMS and CBMS

Knowledge about fractional parts of linear polynomials is fairly
satisfactory. Knowledge about fractional parts of nonlinear polynomials is not
so satisfactory. In these notes the author starts out with Heilbronn's Theorem
on quadratic polynomials and branches out in three directions. In Sections
7–12 he deals with arbitrary polynomials with constant term zero. In
Sections 13–19 he takes up simultaneous approximation of quadratic
polynomials. In Sections 20–21 he discusses special quadratic polynomials
in several variables.

There are many open questions: in fact, most of the results obtained in
these notes ar almost certainly not best possible. Since the theory is not in
its final form including the most general situation, i.e. simultaneous
fractional parts of polynomials in several variables of arbitary degree. On
the other hand, he has given all proofs in full detail and at a leisurely
pace.

For the first half of this work, only the standard notions of an
undergraduate number theory course are required. For the second half, some
knowledge of the geometry of numbers is helpful.

#### Table of Contents

# Table of Contents

## Small Fractional Parts of Polynomials

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents iii4 free
- Preface v6 free
- 1. Heilbronn's Theorem 18 free
- 2. The Heilbronn Alternative Lemma 310
- 3. Vinogradov's Lemma 411
- 4. About Sums Σ||ξ[sub(i)]||[sup(…1)] 512
- 5. About Sums Σe(an2) 613
- 6. Proof of the Heilbronn Alternative Lemma 815
- 7. Fractional Parts of Polynomials 916
- 8. A General Alternative Lemma 1017
- 9. Sums Σ||ξ[sub(i)]||[sup(…1)] Again 1320
- 10. Estimation of Weyl Sums 1522
- 11. What Happens if the Weyl Sums are Large 1825
- 12. Proof of the General Alternative Theorem 2027
- 13. Simultaneous Approximation 2128
- 14. A Reduction 2330
- 15. A Vinogradov Lemma 2633
- 16. Proof of the Alternative Lemma on Simultaneous Approximation 2734
- 17. On max||a[sub(i)n[sup(2)]|| 2936
- 18. A Determinant Argument 3239
- 19. Proof of the Three Alternatives Lemma 3441
- 20. Quadratic Polynomials in Several Variables 3542
- 21. Proofs for Quadratic Polynomials 3643
- References 4047
- Back Cover Back Cover149