Small Fractional Parts of Polynomials, Numéro 32

Couverture
American Mathematical Soc., 1977 - 41 pages
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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.
 

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Table des matières

Heilbronns Theorem
1
The Heilbronn Alternative Lemma
3
Vinogradovs Lemma
4
About Sums ξi1
5
About Sums eαn2
6
Proof of the Heilbronn Alternative Lemma
8
Parts of Polynomials
9
A General Alternative Lemma
10
Proof of the General Alternative Lemma
20
Simultaneous Approximation
21
A Reduction
23
A Vinogradov Lemma
26
Proof of the Alternative Lemma on Simultaneous Approximation
27
On maxαn2
29
A Determinant Argument
32
Proof of the Three Alternatives Lemma
34

Sums ξi1 again
13
Estimation of Weyl Sums
15
What Happens if the Weyl Sums are Large
18
Quadratic Polynomials in Several Variables
35
Proofs for Quadratic Polynomials
36
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