Small Fractional Parts of PolynomialsAmerican Mathematical Soc., 1977 - 41 pages 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 des matières
Heilbronns Theorem | 3 |
The Heilbronn Alternative Lemma | 5 |
Vinogradovs Lemma | 6 |
About Sums ξi1 | 7 |
About Sums eαn2 | 8 |
Proof of the Heilbronn Alternative Lemma | 10 |
Parts of Polynomials | 11 |
A General Alternative Lemma | 12 |
Proof of the General Alternative Lemma | 22 |
Simultaneous Approximation | 23 |
A Reduction | 25 |
A Vinogradov Lemma | 28 |
Proof of the Alternative Lemma on Simultaneous Approximation | 29 |
On maxαn2 | 31 |
A Determinant Argument | 34 |
Proof of the Three Alternatives Lemma | 36 |
Sums ξi1 again | 15 |
Estimation of Weyl Sums | 17 |
What Happens if the Weyl Sums are Large | 20 |
Quadratic Polynomials in Several Variables | 37 |
Proofs for Quadratic Polynomials | 38 |
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Expressions et termes fréquents
a¸n² a₁ a₁n² algebraic alternative holds an² apply Lemma 11A B₂ Bn² c₁ c₁(e c₂ Cauchy's inequality consecutive integers consisting of points constant term zero coprime d(II Davenport 1967 diophantine Dirichlet's Theorem Estimation of Weyl Fourier expansion fractional Heilbronn Alternative Lemma Heilbronn's Theorem interval K₁ lattice points Lectures Lemma 4D Lemma 8A Lemma follows Lemma on Simultaneous linearly independent log q Math min(N Minkowski's modules modulo n*)² qa N₁ n²a natural n natural q NK-k Nº¹ number of divisors obtain P₁ parallelepiped polynomial an³ polynomial f(n Proof Putting q₂ quadratic nonresidue quadratic polynomials ratic real number replaced s-tuples Schmidt Simultaneous Approximation smaller order subinterval successive minima summands Theorem 7B theorem of Heilbronn theory Three Alternatives Lemma variables vectors Vinogradov Vinogradov's Lemma Weyl Sums whence Write y₁ z₁ Σ Σ