Additive combinatorics:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2006
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Cambridge studies in advanced mathematics
105 |
| Schlagworte: | |
| Online-Zugang: | Beschreibung für Leser Inhaltsverzeichnis Inhaltsverzeichnis |
| Beschreibung: | Hier auch später erschienene, unveränderte Nachdrucke |
| Beschreibung: | XVIII, 512 S. |
| ISBN: | 9780521853866 0521853869 |
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents page xi Prologue 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 The probabilistic method The first moment method The second moment method The exponential moment method Correlation inequalities The Lovász local lemma Janson’s inequality Concentration of polynomials Thin bases of higher order Thin Waring bases Appendix: the distribution of the primes 1 2 6 9 19 23 27 33 37 42 45 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Sum set estimates Sum sets Doubling constants Rúzsa distance and additive energy Covering lemmas The Balog-Szemerédi-Gowers theorem Symmetry sets and imbalanced partial sum sets Non-commutative analogs Elementary sum-product estimates 51 54 57 59 69 78 83 92 99 3 Additive geometry 112 113 119 122 3.1 Additive groups 3.2 Progressions 3.3 Convex bodies vii
viii Contents 3.4 The Brunn-Minkowski inequality 3.5 Intersecting a convex set with a lattice 3.6 Progressions and proper progressions 127 130 143 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Fourier-analytic methods Basic theory Lp theory Linear bias Bohr sets Л(р) constants, Вң[g] sets, and dissociated sets The spectrum of an additive set Progressions in sum sets 149 150 156 160 165 172 181 189 5 5.1 5.2 5.3 5.4 5.5 5.6 Inverse sum set theorems Minimal size of sum sets and the e-transform Sum sets in vector spaces Freiman homomorphisms Torsion and torsion-free inverse theorems Universal ambient groups Freiman’s theorem in an arbitrary group 198 198 211 220 227 233 239 6 6.1 6.2 6.3 6.4 6.5 Graph-theoretic methods Basic notions Independent sets, sum-free subsets, and Sidon sets Ramsey theory Proof of the Balog-Szemerédi-Gowers theorem Pliinnecke’s theorem 246 247 248 254 261 267 7 7.1 7.2 7.3 7.4 7.5 7.6 The Littlewood-Offord problem The combinatorial approach The Fourier-analytic approach The Esseen concentration inequality Inverse Littlewood-Offord results Random Bernoulli matrices The quadratic Littlewood-Offord problem 276 277 281 290 292 297 304 8 8.1 8.2 8.3 8.4 8.5 Incidence geometry The crossing number of a graph The Szemerédi-Trotter theorem The sum-product problem in R Cell decompositions and the distinct distances problem The sum-product problem in other fields 308 308 311 315 319 325
Contents 9 Algebraic methods The combinatorial Nullstellensatz ix 329 330 333 342 345 350 354 356 362 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Restricted sum sets Snevily’s conjecture Finite fields Davenport’s problem Kemnitz’s conjecture Stepanov’s method Cyclotomie fields, and the uncertainty principle 10 Szemerédi’s theorem for к = 3 10.1 10.2 10.3 10.4 10.5 10.6 10.7 General strategy The small torsion case The integer case Quantitative bounds An ergodic argument The Szemerédi regularity lemma Szemerédi’s argument 11 Szemerédi’s theorem for к 3 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Gowers uniformity norms Hard obstructions to uniformity Proof of Theorem 11.6 Soft obstructions to uniformity The infinitary ergodic approach The hypergraph approach Arithmetic progressions in the primes 12 Long arithmetic progressions in sum sets 12.1 12.2 12.3 12.4 12.5 12.6 Introduction Proof of Theorem 12.4 Generalizations and variants Complete and subcomplete sequences Proof of Theorem 12.17 Further applications 470 470 473 477 480 482 484 Bibliography Index 488 505 369 372 378 386 389 398 406 411 414 417 424 432 440 448 454 463
|
| adam_txt |
Contents page xi Prologue 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 The probabilistic method The first moment method The second moment method The exponential moment method Correlation inequalities The Lovász local lemma Janson’s inequality Concentration of polynomials Thin bases of higher order Thin Waring bases Appendix: the distribution of the primes 1 2 6 9 19 23 27 33 37 42 45 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Sum set estimates Sum sets Doubling constants Rúzsa distance and additive energy Covering lemmas The Balog-Szemerédi-Gowers theorem Symmetry sets and imbalanced partial sum sets Non-commutative analogs Elementary sum-product estimates 51 54 57 59 69 78 83 92 99 3 Additive geometry 112 113 119 122 3.1 Additive groups 3.2 Progressions 3.3 Convex bodies vii
viii Contents 3.4 The Brunn-Minkowski inequality 3.5 Intersecting a convex set with a lattice 3.6 Progressions and proper progressions 127 130 143 4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Fourier-analytic methods Basic theory Lp theory Linear bias Bohr sets Л(р) constants, Вң[g] sets, and dissociated sets The spectrum of an additive set Progressions in sum sets 149 150 156 160 165 172 181 189 5 5.1 5.2 5.3 5.4 5.5 5.6 Inverse sum set theorems Minimal size of sum sets and the e-transform Sum sets in vector spaces Freiman homomorphisms Torsion and torsion-free inverse theorems Universal ambient groups Freiman’s theorem in an arbitrary group 198 198 211 220 227 233 239 6 6.1 6.2 6.3 6.4 6.5 Graph-theoretic methods Basic notions Independent sets, sum-free subsets, and Sidon sets Ramsey theory Proof of the Balog-Szemerédi-Gowers theorem Pliinnecke’s theorem 246 247 248 254 261 267 7 7.1 7.2 7.3 7.4 7.5 7.6 The Littlewood-Offord problem The combinatorial approach The Fourier-analytic approach The Esseen concentration inequality Inverse Littlewood-Offord results Random Bernoulli matrices The quadratic Littlewood-Offord problem 276 277 281 290 292 297 304 8 8.1 8.2 8.3 8.4 8.5 Incidence geometry The crossing number of a graph The Szemerédi-Trotter theorem The sum-product problem in R Cell decompositions and the distinct distances problem The sum-product problem in other fields 308 308 311 315 319 325
Contents 9 Algebraic methods The combinatorial Nullstellensatz ix 329 330 333 342 345 350 354 356 362 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Restricted sum sets Snevily’s conjecture Finite fields Davenport’s problem Kemnitz’s conjecture Stepanov’s method Cyclotomie fields, and the uncertainty principle 10 Szemerédi’s theorem for к = 3 10.1 10.2 10.3 10.4 10.5 10.6 10.7 General strategy The small torsion case The integer case Quantitative bounds An ergodic argument The Szemerédi regularity lemma Szemerédi’s argument 11 Szemerédi’s theorem for к 3 11.1 11.2 11.3 11.4 11.5 11.6 11.7 Gowers uniformity norms Hard obstructions to uniformity Proof of Theorem 11.6 Soft obstructions to uniformity The infinitary ergodic approach The hypergraph approach Arithmetic progressions in the primes 12 Long arithmetic progressions in sum sets 12.1 12.2 12.3 12.4 12.5 12.6 Introduction Proof of Theorem 12.4 Generalizations and variants Complete and subcomplete sequences Proof of Theorem 12.17 Further applications 470 470 473 477 480 482 484 Bibliography Index 488 505 369 372 378 386 389 398 406 411 414 417 424 432 440 448 454 463 |
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| index_date | 2024-07-02T15:25:58Z |
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| language | English |
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| series2 | Cambridge studies in advanced mathematics |
| spelling | Tao, Terence 1975- Verfasser (DE-588)132190370 aut Additive combinatorics Terence Tao ; Van Vu 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2006 XVIII, 512 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 105 Hier auch später erschienene, unveränderte Nachdrucke Kombinatorik - Additive Zahlentheorie Additive combinatorics Kombinatorik (DE-588)4031824-2 gnd rswk-swf Graphentheorie (DE-588)4113782-6 gnd rswk-swf Kombinatorik (DE-588)4031824-2 s Graphentheorie (DE-588)4113782-6 s DE-604 Vu, Van Verfasser aut Cambridge studies in advanced mathematics 105 (DE-604)BV000003678 105 http://www.loc.gov/catdir/enhancements/fy0618/2006299772-d.html Beschreibung für Leser http://www.loc.gov/catdir/enhancements/fy0618/2006299772-t.html Inhaltsverzeichnis Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014943165&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tao, Terence 1975- Vu, Van Additive combinatorics Cambridge studies in advanced mathematics Kombinatorik - Additive Zahlentheorie Additive combinatorics Kombinatorik (DE-588)4031824-2 gnd Graphentheorie (DE-588)4113782-6 gnd |
| subject_GND | (DE-588)4031824-2 (DE-588)4113782-6 |
| title | Additive combinatorics |
| title_auth | Additive combinatorics |
| title_exact_search | Additive combinatorics |
| title_exact_search_txtP | Additive combinatorics |
| title_full | Additive combinatorics Terence Tao ; Van Vu |
| title_fullStr | Additive combinatorics Terence Tao ; Van Vu |
| title_full_unstemmed | Additive combinatorics Terence Tao ; Van Vu |
| title_short | Additive combinatorics |
| title_sort | additive combinatorics |
| topic | Kombinatorik - Additive Zahlentheorie Additive combinatorics Kombinatorik (DE-588)4031824-2 gnd Graphentheorie (DE-588)4113782-6 gnd |
| topic_facet | Kombinatorik - Additive Zahlentheorie Additive combinatorics Kombinatorik Graphentheorie |
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| volume_link | (DE-604)BV000003678 |
| work_keys_str_mv | AT taoterence additivecombinatorics AT vuvan additivecombinatorics |
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