Additive Combinatorics: A Menu of Research Problems

Content: Cover --
HAlf title --
Series Editors --
Title --
Copyrights --
Contents --
Preface --
Notations --
I Ingredients --
1 Number theory --
1.1 Divisibility of integers --
1.2 Congruences --
1.3 The Fundamental Theorem of Number Theory --
1.4 Multiplicative number theory --
1.5 Additive number theory --
2 Combinatorics --
2.1 Basic enumeration principles --
2.2 Counting lists, sequences, sets, and multisets --
2.3 Binomial coefficients and Pascal\'s Triangle --
2.4 Some recurrence relations --
2.5 The integer lattice and its layers --
3 Group theory --
3.1 Finite abelian groups --
3.2 Group isomorphisms --
3.3 The Fundamental Theorem of Finite Abelian Groups --
3.4 Subgroups and cosets --
3.5 Subgroups generated by subsets --
3.6 Sumsets --
II Appetizers --
Spherical designs --
Caps, centroids, and the game SET --
How many elements does it take to span a group? --
In pursuit of perfection --
The declaration of independence --
III Sides --
The function vg(n, h) --
The function v±(n, h) --
The function u(n,m, h) --
The function uˆ(n,m, h) --
IV Entrees --
A Maximum sumset size --
A.1 Unrestricted sumsets --
A.1.1 Fixed number of terms --
A.1.2 Limited number of terms --
A.1.3 Arbitrary number of terms --
A.2 Unrestricted signed sumsets --
A.2.1 Fixed number of terms --
A.2.2 Limited number of terms --
A.2.3 Arbitrary number of terms --
A.3 Restricted sumsets --
A.3.1 Fixed number of terms --
A.3.2 Limited number of terms --
A.3.3 Arbitrary number of terms --
A.4 Restricted signed sumsets --
A.4.1 Fixed number of terms --
A.4.2 Limited number of terms --
A.4.3 Arbitrary number of terms --
B Spanning sets --
B.1 Unrestricted sumsets --
B.1.1 Fixed number of terms --
B.1.2 Limited number of terms --
B.1.3 Arbitrary number of terms --
B.2 Unrestricted signed sumsets --
B.2.1 Fixed number of terms --
B.2.2 Limited number of terms. B.2.3 Arbitrary number of terms --
B.3 Restricted sumsets --
B.3.1 Fixed number of terms --
B.3.2 Limited number of terms --
B.3.3 Arbitrary number of terms --
B.4 Restricted signed sumsets --
B.4.1 Fixed number of terms --
B.4.2 Limited number of terms --
B.4.3 Arbitrary number of terms --
C Sidon sets --
C.1 Unrestricted sumsets --
C.1.1 Fixed number of terms --
C.1.2 Limited number of terms --
C.1.3 Arbitrary number of terms --
C.2 Unrestricted signed sumsets --
C.2.1 Fixed number of terms --
C.2.2 Limited number of terms --
C.2.3 Arbitrary number of terms --
C.3 Restricted sumsets --
C.3.1 Fixed number of terms --
C.3.2 Limited number of terms --
C.3.3 Arbitrary number of terms --
C.4 Restricted signed sumsets --
C.4.1 Fixed number of terms --
C.4.2 Limited number of terms --
C.4.3 Arbitrary number of terms --
D Minimum sumset size --
D.1 Unrestricted sumsets --
D.1.1 Fixed number of terms --
D.1.2 Limited number of terms --
D.1.3 Arbitrary number of terms --
D.2 Unrestricted signed sumsets --
D.2.1 Fixed number of terms --
D.2.2 Limited number of terms --
D.2.3 Arbitrary number of terms --
D.3 Restricted sumsets --
D.3.1 Fixed number of terms --
D.3.2 Limited number of terms --
D.3.3 Arbitrary number of terms --
D.4 Restricted signed sumsets --
D.4.1 Fixed number of terms --
D.4.2 Limited number of terms --
D.4.3 Arbitrary number of terms --
E The critical number --
E.1 Unrestricted sumsets --
E.1.1 Fixed number of terms --
E.1.2 Limited number of terms --
E.1.3 Arbitrary number of terms --
E.2 Unrestricted signed sumsets --
E.2.1 Fixed number of terms --
E.2.2 Limited number of terms --
E.2.3 Arbitrary number of terms --
E.3 Restricted sumsets --
E.3.1 Fixed number of terms --
E.3.2 Limited number of terms --
E.3.3 Arbitrary number of terms --
E.4 Restricted signed sumsets --
E.4.1 Fixed number of terms --
E.4.2 Limited number of terms. E.4.3 Arbitrary number of terms --
F Zero-sum-free sets --
F.1 Unrestricted sumsets --
F.1.1 Fixed number of terms --
F.1.2 Limited number of terms --
F.1.3 Arbitrary number of terms --
F.2 Unrestricted signed sumsets --
F.2.1 Fixed number of terms --
F.2.2 Limited number of terms --
F.2.3 Arbitrary number of terms --
F.3 Restricted sumsets --
F.3.1 Fixed number of terms --
F.3.2 Limited number of terms --
F.3.3 Arbitrary number of terms --
F.4 Restricted signed sumsets --
F.4.1 Fixed number of terms --
F.4.2 Limited number of terms --
F.4.3 Arbitrary number of terms --
G Sum-free sets --
G.1 Unrestricted sumsets --
G.1.1 Fixed number of terms --
G.1.2 Limited number of terms --
G.1.3 Arbitrary number of terms --
G.2 Unrestricted signed sumsets --
G.2.1 Fixed number of terms --
G.2.2 Limited number of terms --
G.2.3 Arbitrary number of terms --
G.3 Restricted sumsets --
G.3.1 Fixed number of terms --
G.3.2 Limited number of terms --
G.3.3 Arbitrary number of terms --
G.4 Restricted signed sumsets --
G.4.1 Fixed number of terms --
G.4.2 Limited number of terms --
G.4.3 Arbitrary number of terms --
V Pudding --
Proof of Proposition 2.2 --
Proof of Proposition 3.1 --
Proof of Proposition 3.4 --
Proof of Proposition 3.5 --
Proof of Proposition 4.2 --
Proof of Proposition 4.3 --
Proof of Theorem 4.4 --
Proof of Proposition 4.9 --
Proof of Proposition 4.10 --
Proof of Theorem 4.17 --
Proof of Proposition 4.22 --
Proof of Proposition 4.23 --
Proof of Proposition 4.26 --
Proof of Proposition 4.29 --
Proof of Proposition A.42 --
Proof of Theorem B.8 --
Proof of Proposition B.28 --
Proof of Proposition B.46 --
Proof of Proposition B.54 --
Proof of Proposition B.57 --
Proof of Proposition C.36 --
Proof of Proposition C.50 --
Proof of Proposition C.51 --
Proof of Proposition D.6 --
Proof of Theorem D.8 --
Proof of Theorem D.9 --
Proof of Theorem D.10. Proof of Theorem D.40 --
Proof of Proposition D.42 --
Proof of Theorem D.47 --
Proof of Proposition D.59 --
Proof of Theorem D.72 --
Proof of Proposition D.128 --
Proof of Theorem E.15 --
Proof of Proposition E.76 --
Proof of Lemma E.87 --
Proof of Theorem E.100 --
Proof of Theorem E.108 --
Proof of Theorem E.109 --
Proof of Theorem F.6 --
Proof of Proposition F.27 --
Proof of Proposition F.28 --
Proof of Proposition F.32 --
Proof of Proposition F.35 --
Proof of Proposition F.46 --
Proof of Proposition F.80 --
Proof of Proposition F.83 --
Proof of Theorem F.88 --
Proof of Proposition F.116 --
Proof of Proposition F.156 --
Proof of Proposition F.179 --
Proof of Proposition G.22 --
Proof of Theorem G.27 --
Proof of Proposition G.64 --
Proof of Corollary G.65 --
Proof of Theorem G.67 --
Proof of Proposition G.73 --
Proof of Proposition G.74 --
Proof of Proposition G.82 --
Bibliography --
Author Index.