Muutke küpsiste eelistusi

Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting 2014 ed. [Kõva köide]

  • Formaat: Hardback, 487 pages, kõrgus x laius: 235x155 mm, kaal: 8749 g, 22 Illustrations, black and white; XVI, 487 p. 22 illus., 1 Hardback
  • Sari: Springer Monographs in Mathematics
  • Ilmumisaeg: 22-Oct-2014
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319107402
  • ISBN-13: 9783319107400
Teised raamatud teemal:
  • Kõva köide
  • Hind: 95,02 €*
  • * hind on lõplik, st. muud allahindlused enam ei rakendu
  • Tavahind: 111,79 €
  • Säästad 15%
  • Raamatu kohalejõudmiseks kirjastusest kulub orienteeruvalt 2-4 nädalat
  • Kogus:
  • Lisa ostukorvi
  • Tasuta tarne
  • Tellimisaeg 2-4 nädalat
  • Lisa soovinimekirja
Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting 2014 ed.Suurem pilt
  • Formaat: Hardback, 487 pages, kõrgus x laius: 235x155 mm, kaal: 8749 g, 22 Illustrations, black and white; XVI, 487 p. 22 illus., 1 Hardback
  • Sari: Springer Monographs in Mathematics
  • Ilmumisaeg: 22-Oct-2014
  • Kirjastus: Springer International Publishing AG
  • ISBN-10: 3319107402
  • ISBN-13: 9783319107400
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9783319107400.html
Märksõnad:

This book gives a comprehensive treatment of random phenomena and distribution results in diophantine approximation, with a particular emphasis on quadratic irrationals. It covers classical material on the subject as well as many new results developed by the author over the past decade. A range of ideas from other areas of mathematics are brought to bear with surprising connections to topics such as formulae for class numbers, special values of L-functions, and Dedekind sums. Care is taken to elaborate difficult proofs by motivating major steps and accompanying them with background explanations, enabling the reader to learn the theory and relevant techniques.

Written by one of the acknowledged experts in the field, Probabilistic Diophantine Approximation is presented in a clear and informal style with sufficient detail to appeal to both advanced students and researchers in number theory.

Arvustused

The author has written a monograph with new results in a new direction of research in probability and number theory, including algebraic number theory. The book is presented in a clear style and will be valuable for researchers in the field. (Oto Strauch, Mathematical Reviews, October, 2015)

Novel, important, intuitively appealing mathematics and a highly readable account. Summing Up: Highly recommended. Upper-division undergraduates and above. (D. V. Feldman, Choice, Vol. 52 (11), July, 2015)

Becks book brings together probability theory, algebraic number theory, and combinatorics to forge new ground. the book is accessible to a wide range of mathematicians. Beck did a nice job of motivating the questions and explaining theanswers. (Darren Glass, MAA Reviews, July, 2015)

Part I Global Aspects Randomness of the Irrational Rotation
1 What Is “r;Probabilistic”r; Diophantine Approximation?
3(76)
1.1 The Giant Leap in Uniform Distribution
3(14)
1.1.1 From Quasi-Periodicity to Randomness
14(1)
1.1.2 Summary in a Nutshell
14(3)
1.2 Randomness in Lattice Point Counting
17(12)
1.2.1 A Key Tool: Ostrowski's Explicit Formula
23(3)
1.2.2 Counting Lattice Points in General
26(3)
1.3 First Warm-Up: Van der Corput Sequence---When Independence Is Given
29(15)
1.3.1 Digit Sums and Generalized Digit Sums
37(1)
1.3.2 A Decomposition Trick
38(5)
1.3.3 Concluding Remark
43(1)
1.4 Second Warm-Up: Markov Chains and the Area Principle
44(15)
1.4.1 Statistical Independence and Markov Chains
49(3)
1.4.2 Long Runs of Heads
52(7)
1.5 The Golden Ratio and Markov Chains: The Simplest Case of Theorem 1.2
59(20)
1.5.1 Constructing the Underlying (Homogeneous) Markov Chain
65(4)
1.5.2 How to Approximate with a Sum of Independent Random Variables
69(1)
1.5.3 Solving the Parity Problem
70(7)
1.5.4 Concluding Remarks
77(2)
2 Expectation, and Its Connection with Quadratic Fields
79(88)
2.1 Computing the Expectation in General (I)
79(21)
2.1.1 An Important Detour: How to Guess Proposition 2.1?
82(1)
2.1.2 Quadratic Fields in a Nutshell
83(4)
2.1.3 Another Detour: Formulating a “r;Positivity Conjecture”r;
87(11)
2.1.4 Proposition 2.1 and Some Works of Hardy and Littlewood
98(2)
2.2 Computing the Expectation in General (II)
100(16)
2.2.1 The Expectation in Theorem 1.1
100(5)
2.2.2 An Analog of Proposition 2.1
105(8)
2.2.3 Periodicity in Proposition 2.9
113(3)
2.3 Fourier Series and a Problem of Hardy and Littlewood (I)
116(12)
2.3.1 Badly Approximable Numbers
118(2)
2.3.2 The Hardy--Littlewood Series
120(3)
2.3.3 Doubling and Halving in Continued Fractions
123(2)
2.3.4 A Geometric Interpretation
125(3)
2.4 Fourier Series and a Problem of Hardy and Littlewood (II)
128(9)
2.5 A Detour: The Giant Leap in Number Theory
137(11)
2.5.1 Looking at the “r;Big Picture”r;
137(11)
2.6 Connection with Quadratic Fields (I)
148(19)
2.6.1 A Detour: Another Class Number Formula
161(2)
2.6.2 How to Compute the Class Number in General: The Complex Case
163(4)
3 Variance, and Its Connection with Quadratic Fields
167(40)
3.1 Computing the Variance
167(9)
3.1.1 Guiding Intuition
168(2)
3.1.2 An Alternative Form of the Guiding Intuition
170(6)
3.2 Connection with Quadratic Fields (II)
176(20)
3.2.1 A Convenient Special Case: When the Class Number Is One
181(1)
3.2.2 The Class Number for Real Quadratic Fields: Illustrations
182(4)
3.2.3 The Dedekind's Zeta Function at s=2: A Formula Involving Characters
186(6)
3.2.4 An Alternative Formula Due to Siegel: Proposition 3.7
192(4)
3.3 Connection with Quadratic Fields (III)
196(11)
3.3.1 The General Case: Computing the Variance for an Arbitrary Quadratic Irrational
196(1)
3.3.2 Computing the Variance in Theorem 1.1: A Special Case
197(5)
3.3.3 Computing the Variance in Theorem 1.1: The General Case
202(2)
3.3.4 The Case of Symmetric Intervals
204(3)
4 Proving Randomness
207(44)
4.1 Completing the Proof of Theorem 1.2
207(16)
4.1.1 Renewal Versus Self-Similarity
210(10)
4.1.2 Ergodic Markov Chains: Exponentially Fast Convergence to the Stationary Distribution
220(3)
4.2 How to Use Lemma 4.2 to Find the Analog of (1.223) in General?
223(3)
4.3 Completing the Proof of Theorem 1.1
226(1)
4.4 The Fourier Series Approach
226(14)
4.4.1 Guiding Intuition
227(6)
4.4.2 Constructing a Sum X1 + X2 + X3 + ...of Almost Independent Random Variables
233(3)
4.4.3 Defining the Truly Independent Random Variables X1, X2, X3
236(4)
4.5 More Results in a Nutshell
240(11)
Part II Local Aspects Inhomogeneous Pell Inequalities
5 Pell's Equation, Superirregularity and Randomness
251(120)
5.1 From Pell Equation to Superirregularity
251(12)
5.1.1 Pell's Equation: Bounded Fluctuations
251(2)
5.1.2 The Area Principle
253(3)
5.1.3 The Giant Leap in the Inhomogeneous Case: Extra Large Fluctuations
256(7)
5.2 Randomness and the Area Principle
263(12)
5.3 Proving Theorem 5.3 and the Lemmas
275(6)
5.4 The Riesz Product
281(7)
5.4.1 The Method of Nested Intervals vs. the Riesz Product
281(4)
5.4.2 The “r;Rectangle Property”r;, and a Key Result: Theorem 5.11
285(3)
5.5 Starting the Proof of Theorem 5.11 Using Riesz Product
288(14)
5.5.1 What are the Trivial Errors and How to Synchronize Them
295(1)
5.5.2 Geometric Ideas
296(3)
5.5.3 An Important Consequence of the “r;Rectangle Property”r;
299(1)
5.5.4 Choosing a Short Vertical Translation
300(1)
5.5.5 Summarizing the Vague Geometric Intuition
301(1)
5.6 More on the Riesz Product
302(12)
5.6.1 Applying Super-Orthogonality
302(5)
5.6.2 Single Term Domination: Clarifying the Technical Details
307(4)
5.6.3 A Combination of the Rectangle Property and the Pigeonhole Principle
311(3)
5.7 Completing the Case Study
314(17)
5.7.1 Verifying (5.152)
314(4)
5.7.2 A Combination of the Rectangle Property and the Pigeonhole Principle
318(6)
5.7.3 A Combination of the Rectangle Property and the Pigeonhole Principle
324(5)
5.7.4 A Combination of the Rectangle Property and the Pigeonhole Principle
329(2)
5.8 Completing the Proof of Theorem 5.11
331(7)
5.9 Yet Another Generalization of Theorem 5.3
338(11)
5.9.1 Step One
341(2)
5.9.2 Step Two: Small “r;Digit”r; ai Implies “r;Local”r; Rectangle Property
343(3)
5.9.3 Step Three: Employing the Riesz Product Technique
346(1)
5.9.4 Step Four: Constructing a Cantor Set
347(2)
5.10 General Point Sets: Theorem 5.19
349(8)
5.10.1 Statistical Version of the Rectangle Property: An Average Argument
351(2)
5.10.2 Consequences of Inequality (5.327)
353(4)
5.11 The Area Principle in General
357(14)
6 More on Randomness
371(110)
6.1 Starting the Proof of Theorem 5.4: Blocks-and-Gaps Decomposition
371(12)
6.2 Completing the Blocks-and-Gaps Decomposition
383(10)
6.3 Estimating the Variance
393(10)
6.4 Applying Probability Theory
403(10)
6.4.1 Central Limit Theorem with Explicit Error Term
405(8)
6.5 Conclusion of the Proof of Theorem 5.4
413(10)
6.6 Proving the Three Lemmas: Part One
423(11)
6.6.1 Properties of the Auxiliary Functions in (6.222) and (6.223)
427(2)
6.6.2 Deduction of Lemma 6.6 from Lemmas 6.4 and 6.5
429(5)
6.7 Proving the Three Lemmas: Part Two
434(12)
6.8 Starting the Proof of Theorem 5.6
446(11)
6.9 Completing the Proof of Theorem 5.6
457(11)
6.10 More Results in a Nutshell
468(13)
6.10.1 Combining the Logarithmic Density with the Central Limit Theorem
473(8)
References 481(4)
Index 485
József Beck is Harold H. Martin Professor of Mathematics at Rutgers University.