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Additive Combinatorics: A Menu of Research Problems [Kõva köide]

(Gettysburg College, USA)
  • Formaat: Hardback, 390 pages, kõrgus x laius: 254x178 mm, kaal: 900 g, 46 Tables, black and white; 10 Illustrations, black and white
  • Sari: Discrete Mathematics and Its Applications
  • Ilmumisaeg: 23-Apr-2018
  • Kirjastus: CRC Press Inc
  • ISBN-10: 0815353014
  • ISBN-13: 9780815353010
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Additive Combinatorics: A Menu of Research ProblemsSuurem pilt
  • Formaat: Hardback, 390 pages, kõrgus x laius: 254x178 mm, kaal: 900 g, 46 Tables, black and white; 10 Illustrations, black and white
  • Sari: Discrete Mathematics and Its Applications
  • Ilmumisaeg: 23-Apr-2018
  • Kirjastus: CRC Press Inc
  • ISBN-10: 0815353014
  • ISBN-13: 9780815353010
Teised raamatud teemal:
Teised raamatud sellest sooduspakkumisest: Taylor & Francis teadusraamatud International Student Editions hindadega
Püsilink: https://www.kriso.ee/db/9780815353010.html
Märksõnad:
Additive Combinatorics: A Menu of Research Problems is the first book of its kind to provide readers with an opportunity to actively explore the relatively new field of additive combinatorics. The author has written the book specifically for students of any background and proficiency level, from beginners to advanced researchers. It features an extensive menu of research projects that are challenging and engaging at many different levels. The questions are new and unsolved, incrementally attainable, and designed to be approachable with various methods.

The book is divided into five parts which are compared to a meal. The first part is called Ingredients and includes relevant background information about number theory, combinatorics, and group theory. The second part, Appetizers, introduces readers to the books main subject through samples. The third part, Sides, covers auxiliary functions that appear throughout different chapters. The books main course, so to speak, is Entrees: it thoroughly investigates a large variety of questions in additive combinatorics by discussing what is already known about them and what remains unsolved. These include maximum and minimum sumset size, spanning sets, critical numbers, and so on. The final part is Pudding and features numerous proofs and results, many of which have never been published.

Features:











The first book of its kind to explore the subject





Students of any level can use the book as the basis for research projects





The text moves gradually through five distinct parts, which is suitable both for beginners without prerequisites and for more advanced students





Includes extensive proofs of propositions and theorems





Each of the introductory chapters contains numerous exercises to help readers
Preface xi
Notations xv
I Ingredients
1(46)
1 Number theory
5(10)
1.1 Divisibility of integers
5(2)
1.2 Congruences
7(1)
1.3 The Fundamental Theorem of Number Theory
8(1)
1.4 Multiplicative number theory
9(2)
1.5 Additive number theory
11(4)
2 Combinatorics
15(16)
2.1 Basic enumeration principles
15(2)
2.2 Counting lists, sequences, sets, and multisets
17(4)
2.3 Binomial coefficients and Pascal's Triangle
21(2)
2.4 Some recurrence relations
23(3)
2.5 The integer lattice and its layers
26(5)
3 Group theory
31(16)
3.1 Finite abelian groups
32(1)
3.2 Group isomorphisms
33(1)
3.3 The Fundamental Theorem of Finite Abelian Groups
34(2)
3.4 Subgroups and cosets
36(3)
3.5 Subgroups generated by subsets
39(1)
3.6 Sumsets
40(7)
II Appetizers
47(26)
Spherical designs
50(6)
Caps, centroids, and the game SET
56(6)
How many elements does it take to span a group?
62(3)
In pursuit of perfection
65(4)
The declaration of independence
69(4)
III Sides
73(20)
The function υg(n, h)
76(5)
The function υ±(n, h)
81(2)
The function υ (n, m, h)
83(3)
The function υˆ(n, m, h)
86(7)
IV Entrees
93(212)
A Maximum sumset size
97(14)
A.1 Unrestricted sumsets
97(1)
A.1.1 Fixed number of terms
98(1)
A.1.2 Limited number of terms
99(1)
A.1.3 Arbitrary number of terms
100(1)
A.2 Unrestricted signed sumsets
101(1)
A.2.1 Fixed number of terms
101(2)
A.2.2 Limited number of terms
103(1)
A.2.3 Arbitrary number of terms
104(1)
A.3 Restricted sumsets
104(1)
A.3.1 Fixed number of terms
104(1)
A.3.2 Limited number of terms
105(1)
A.3.3 Arbitrary number of terms
106(1)
A.4 Restricted signed sumsets
107(1)
A.4.1 Fixed number of terms
107(1)
A.4.2 Limited number of terms
108(1)
A.4.3 Arbitrary number of terms
109(2)
B Spanning sets
111(20)
B.1 Unrestricted sumsets
112(1)
B.1.1 Fixed number of terms
112(1)
B.1.2 Limited number of terms
112(7)
B.1.3 Arbitrary number of terms
119(1)
B.2 Unrestricted signed sumsets
119(1)
B.2.1 Fixed number of terms
119(3)
B.2.2 Limited number of terms
122(5)
B.2.3 Arbitrary number of terms
127(1)
B.3 Restricted sumsets
127(1)
B.3.1 Fixed number of terms
127(1)
B.3.2 Limited number of terms
127(2)
B.3.3 Arbitrary number of terms
129(1)
B.4 Restricted signed sumsets
129(1)
B.4.1 Fixed number of terms
129(1)
B.4.2 Limited number of terms
129(1)
B.4.3 Arbitrary number of terms
129(2)
C Sidon sets
131(18)
C.1 Unrestricted sumsets
132(1)
C.1.1 Fixed number of terms
132(7)
C.1.2 Limited number of terms
139(1)
C.1.3 Arbitrary number of terms
139(1)
C.2 Unrestricted signed sumsets
139(1)
C.2.1 Fixed number of terms
140(2)
C.2.2 Limited number of terms
142(1)
C.2.3 Arbitrary number of terms
143(1)
C.3 Restricted sumsets
143(1)
C.3.1 Fixed number of terms
143(2)
C.3.2 Limited number of terms
145(2)
C.3.3 Arbitrary number of terms
147(1)
C.4 Restricted signed sumsets
147(1)
C.4.1 Fixed number of terms
147(1)
C.4.2 Limited number of terms
147(1)
C.4.3 Arbitrary number of terms
147(2)
D Minimum sumset size
149(44)
D.1 Unrestricted sumsets
149(1)
D.1.1 Fixed number of terms
149(6)
D.1.2 Limited number of terms
155(1)
D.1.3 Arbitrary number of terms
156(1)
D.2 Unrestricted signed sumsets
156(1)
D.2.1 Fixed number of terms
156(5)
D.2.2 Limited number of terms
161(2)
D.2.3 Arbitrary number of terms
163(1)
D.3 Restricted sumsets
163(1)
D.3.1 Fixed number of terms
163(20)
D.3.2 Limited number of terms
183(1)
D.3.3 Arbitrary number of terms
184(7)
D.4 Restricted signed sumsets
191(1)
D.4.1 Fixed number of terms
191(1)
D.4.2 Limited number of terms
191(1)
D.4.3 Arbitrary number of terms
191(2)
E The critical number
193(32)
E.1 Unrestricted sumsets
193(1)
E.1.1 Fixed number of terms
193(3)
E.1.2 Limited number of terms
196(5)
E.1.3 Arbitrary number of terms
201(1)
E.2 Unrestricted signed sumsets
202(1)
E.2.1 Fixed number of terms
202(1)
E.2.2 Limited number of terms
203(4)
E.2.3 Arbitrary number of terms
207(1)
E.3 Restricted sumsets
207(1)
E.3.1 Fixed number of terms
207(5)
E.3.2 Limited number of terms
212(2)
E.3.3 Arbitrary number of terms
214(9)
E.4 Restricted signed sumsets
223(1)
E.4.1 Fixed number of terms
223(1)
E.4.2 Limited number of terms
223(1)
E.4.3 Arbitrary number of terms
223(2)
F Zero-sum-free sets
225(50)
F.1 Unrestricted sumsets
225(1)
F.1.1 Fixed number of terms
226(4)
F.1.2 Limited number of terms
230(1)
F.1.3 Arbitrary number of terms
231(1)
F.2 Unrestricted signed sumsets
231(1)
F.2.1 Fixed number of terms
231(4)
F.2.2 Limited number of terms
235(10)
F.2.3 Arbitrary number of terms
245(1)
F.3 Restricted sumsets
246(1)
F.3.1 Fixed number of terms
246(12)
F.3.2 Limited number of terms
258(3)
F.3.3 Arbitrary number of terms
261(5)
F.4 Restricted signed sumsets
266(1)
F.4.1 Fixed number of terms
266(3)
F.4.2 Limited number of terms
269(2)
F.4.3 Arbitrary number of terms
271(4)
G Sum-free sets
275(30)
G.1 Unrestricted sumsets
276(1)
G.1.1 Fixed number of terms
276(14)
G.1.2 Limited number of terms
290(3)
G.1.3 Arbitrary number of terms
293(1)
G.2 Unrestricted signed sumsets
293(1)
G.2.1 Fixed number of terms
293(1)
G.2.2 Limited number of terms
293(1)
G.2.3 Arbitrary number of terms
293(1)
G.3 Restricted sumsets
294(1)
G.3.1 Fixed number of terms
294(9)
G.3.2 Limited number of terms
303(1)
G.3.3 Arbitrary number of terms
303(1)
G.4 Restricted signed sumsets
303(1)
G.4.1 Fixed number of terms
303(1)
G.4.2 Limited number of terms
303(1)
G.4.3 Arbitrary number of terms
303(2)
V Pudding
305(72)
Proof of Proposition 2.2
308(1)
Proof of Proposition 3.1
309(1)
Proof of Proposition 3.4
309(2)
Proof of Proposition 3.5
311(1)
Proof of Proposition 4.2
311(2)
Proof of Proposition 4.3
313(1)
Proof of Theorem 4.4
314(1)
Proof of Proposition 4.9
315(2)
Proof of Proposition 4.10
317(1)
Proof of Theorem 4.17
318(1)
Proof of Proposition 4.22
318(1)
Proof of Proposition 4.23
319(1)
Proof of Proposition 4.26
320(1)
Proof of Proposition 4.29
321(1)
Proof of Proposition A.42
322(1)
Proof of Theorem B.8
323(2)
Proof of Proposition B.28
325(1)
Proof of Proposition B.46
325(2)
Proof of Proposition B.54
327(1)
Proof of Proposition B.57
327(2)
Proof of Proposition C.36
329(1)
Proof of Proposition C.50
330(1)
Proof of Proposition C.51
331(1)
Proof of Proposition D.6
332(1)
Proof of Theorem D.8
333(1)
Proof of Theorem D.9
334(1)
Proof of Theorem D.10
335(1)
Proof of Theorem D.40
335(1)
Proof of Proposition D.42
336(1)
Proof of Theorem D.47
337(7)
Proof of Proposition D.59
344(2)
Proof of Theorem D.72
346(2)
Proof of Proposition D.128
348(1)
Proof of Theorem E.15
349(1)
Proof of Proposition E.76
350(1)
Proof of Lemma E.87
350(1)
Proof of Theorem E.100
351(1)
Proof of Theorem E.108
351(2)
Proof of Theorem E.109
353(1)
Proof of Theorem F.6
354(1)
Proof of Proposition F.27
355(1)
Proof of Proposition F.28
356(1)
Proof of Proposition F.32
356(1)
Proof of Proposition F.35
357(1)
Proof of Proposition F.46
358(2)
Proof of Proposition F.80
360(1)
Proof of Proposition F.83
361(1)
Proof of Theorem F.88
362(1)
Proof of Proposition F.116
363(1)
Proof of Proposition F. 156
363(2)
Proof of Proposition F.179
365(1)
Proof of Proposition G.22
365(3)
Proof of Theorem G.27
368(1)
Proof of Proposition G.64
369(1)
Proof of Corollary G.65
370(2)
Proof of Theorem G.67
372(1)
Proof of Proposition G.73
373(1)
Proof of Proposition G.74
373(1)
Proof of Proposition G.82
374(3)
Bibliography 377(11)
Author Index 388
Béla Bajnok is a Professor of Mathematics at Gettysburg College and an endowed Alumni Chair. He holds a Ph.D. from Ohio State University and has won several teaching awards, including the Creative Teaching Award at Gettysburg College and the Crawford Teaching Award from the Mathematical Association of America.