Additive Combinatorics. A Menu of Research Problems

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Contents......Page 3 Preface......Page 8 Notation......Page 11 --- Ingredients......Page 16 Divisibility of integers......Page 18 Congruences......Page 20 The Fundamental Theorem of Number Theory......Page 21 Multiplicative number theory......Page 22 Additive number theory......Page 24 Basic enumeration principles......Page 28 Counting lists, sequences, sets, and multisets......Page 30 Binomial coefficients and Pascal’s Triangle......Page 34 Some recurrence relations......Page 36 The integer lattice and its layers......Page 39 Group theory......Page 43 Finite abelian groups......Page 44 Group isomorphisms......Page 45 The Fundamental Theorem of Finite Abelian Groups......Page 46 Subgroups and cosets......Page 48 Subgroups generated by subsets......Page 51 Sumsets......Page 52 Appetizers......Page 59 Spherical designs......Page 61 Caps, centroids, and the game SET......Page 67 How many elements does it take to span a group?......Page 73 In pursuit of perfection......Page 76 The declaration of independence......Page 80 Sides......Page 84 The function......Page 86 --- Entrees......Page 102 A.1 Unrestricted sumsets......Page 104 A.2 Unrestricted signed sumsets......Page 108 A.3 Restricted sumsets......Page 111 A.4 Restricted signed sumsets......Page 114 Spanning sets......Page 118 B.1 Unrestricted sumsets......Page 119 B.2 Unrestricted signed sumsets......Page 126 B.3 Restricted sumsets......Page 134 B.4 Restricted signed sumsets......Page 136 Sidon sets......Page 137 C.1 Unrestricted sumsets......Page 138 C.2 Unrestricted signed sumsets......Page 145 C.3 Restricted sumsets......Page 149 C.4 Restricted signed sumsets......Page 153 D.1 Unrestricted sumsets......Page 155 D.2 Unrestricted signed sumsets......Page 162 D.3 Restricted sumsets......Page 169 D.4 Restricted signed sumsets......Page 197 E.1 Unrestricted sumsets......Page 198 E.2 Unrestricted signed sumsets......Page 207 E.3 Restricted sumsets......Page 212 E.4 Restricted signed sumsets......Page 228 F.1 Unrestricted sumsets......Page 229 F.2 Unrestricted signed sumsets......Page 235 F.3 Restricted sumsets......Page 250 F.4 Restricted signed sumsets......Page 270 Sum-free sets......Page 279 G.1 Unrestricted sumsets......Page 280 G.2 Unrestricted signed sumsets......Page 297 G.3 Restricted sumsets......Page 298 G.4 Restricted signed sumsets......Page 307 Pudding......Page 308 Proof of Proposition 2.2......Page 310 Proof of Proposition 3.1......Page 311 Proof of Proposition 3.4......Page 312 Proof of Proposition 3.5......Page 313 Proof of Proposition 4.2......Page 314 Proof of Proposition 4.3......Page 315 Proof of Theorem 4.4......Page 316 Proof of Proposition 4.9......Page 318 Proof of Proposition 4.10......Page 319 Proof of Theorem 4.17......Page 320 Proof of Proposition 4.22......Page 321 Proof of Proposition 4.26......Page 322 Proof of Proposition 4.29......Page 323 Proof of Theorem B.8......Page 325 Proof of Proposition B.46......Page 327 Proof of Proposition B.57......Page 329 Proof of Proposition C.36......Page 331 Proof of Proposition C.50......Page 332 Proof of Proposition C.51......Page 333 Proof of Theorem D.8......Page 335 Proof of Theorem D.9......Page 336 Proof of Theorem D.40......Page 337 Proof of Proposition D.42......Page 338 Proof of Theorem D.47......Page 339 Proof of Proposition D.59......Page 346 Proof of Theorem D.72......Page 348 Proof of Theorem E.15......Page 351 Proof of Proposition E.76......Page 352 Proof of Theorem E.100......Page 353 Proof of Theorem E.108......Page 354 Proof of Theorem E.109......Page 355 Proof of Theorem F.6......Page 356 Proof of Proposition F.27......Page 357 Proof of Proposition F.28......Page 358 Proof of Proposition F.35......Page 359 Proof of Proposition F.46......Page 360 Proof of Proposition F.80......Page 362 Proof of Proposition F.83......Page 363 Proof of Theorem F.88......Page 364 Proof of Proposition F.156......Page 365 Proof of Proposition F.179......Page 367 Proof of Proposition G.22......Page 368 Proof of Theorem G.27......Page 370 Proof of Proposition G.64......Page 371 Proof of Corollary G.65......Page 372 Proof of Theorem G.67......Page 374 Proof of Proposition G.73......Page 375 Proof of Proposition G.82......Page 376 Biblio......Page 378