Area, Lattice Points, and Exponential Sums

Cover......Page 1
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Contents......Page 8
Notation......Page 12
Introduction......Page 14
Part I Elementary methods......Page 18
1.1 Height......Page 20
1.2 The Farey sequence......Page 21
1.3 Lattices and the modular group......Page 24
1.4 Uniform distribution......Page 25
1.5 Approximating a given real number......Page 28
1.6 Uniform Diophantine approximation......Page 32
2.1 Counting squares......Page 37
2.2 Jarnik\'s polygon......Page 44
2.3 The discrepancy of a polygon......Page 46
2.4 Fitting a polygon to a smooth curve......Page 51
3.1 Introduction......Page 55
3.2 The reduction step......Page 58
3.3 The iteration......Page 61
3.4 Swinnerton-Dyer\'s method: the convex polygon......Page 67
3.5 Swinnerton-Dyer\'s method: counting quadruplets......Page 69
3.6 Expanding the result......Page 73
3.7 Points on the curve......Page 75
4.1 Major and minor sides of the polygon......Page 76
4.2 Duality......Page 85
4.3 A linear form often near an integer......Page 94
Part II The Bombieri-Iwaniec method......Page 98
5.1 Bounds for exponential integrals......Page 100
5.2 Partial summation for exponential sums......Page 102
5.3 Rounding error sums......Page 106
5.4 Poisson summation......Page 111
5.5 Evaluating exponential integrals......Page 117
5.6 Bilinear and mean square bounds......Page 127
6.1 The simple exponential sum......Page 139
6.2 The lattice point discrepancy......Page 141
6.3 The mean square discrepancy......Page 148
7.1 Motivation......Page 155
7.2 Major and minor arcs......Page 157
7.3 Poisson summation......Page 162
7.4 The large sieve on the minor arcs......Page 165
7.5 A preliminary calculation......Page 171
7.6 Long major arcs......Page 173
8.1 Major and minor arcs......Page 180
8.2 The major arcs estimate......Page 183
8.3 Poisson summation on the minor arcs......Page 185
8.4 The large sieve on the minor arcs......Page 191
9.1 Major and minor arcs......Page 198
9.2 The major arcs estimate......Page 200
9.3 Poisson summation on the minor arcs......Page 201
9.4 The large sieve on the minor arcs......Page 206
10.1 Modular forms......Page 210
10.2 The Wilton summation formula......Page 213
10.3 Farey arcs......Page 221
10.4 Wilton summation on the Farey arcs......Page 223
10.5 A large sieve on the Farey arcs......Page 226
10.6 Towards a mean square result......Page 233
10.7 Jutila\'s third method......Page 238
Part III The First Spacing Problem: `integer\' vectors......Page 246
11.1 Preparation: divisor functions......Page 248
11.2 Families of solutions......Page 254
11.3 A comparison argument......Page 262
12.1 Integrals that count......Page 268
12.2 The minor arcs......Page 272
12.3 The major arcs......Page 275
12.4 Extrapolation......Page 283
13.1 Families of solutions......Page 285
13.2 Good and bad families......Page 291
13.3 The problem with a perturbing term......Page 296
Part IV The Second Spacing Problem: `rational\' vectors......Page 298
14.1 The Coincidence Conditions......Page 300
14.2 Magic matrices......Page 302
14.3 The Second Condition......Page 308
14.4 A family of sums......Page 313
15.1 Parametrizing rational points......Page 320
15.2 Coincidence over a short interval......Page 324
15.3 Linearizing the Fourth Condition......Page 328
15.4 Coincidence over a long interval......Page 333
15.5 Extending the Taylor series......Page 336
16.1 Counting coincident pairs of minor arcs......Page 342
16.2 Sums with congruence conditions......Page 349
16.3 Eliminating the centres of the arcs......Page 352
Part V Results and applications......Page 360
17.1 The simple exponential sum......Page 362
17.2 Exponential sums with a parameter......Page 370
17.3 A congruence family of sums......Page 374
17.4 Sums with T large......Page 376
18.1 Exponential sums......Page 385
18.2 Integer points and rounding error......Page 390
18.3 Lattice points inside a closed curve......Page 397
18.4 A family of lattice point problems......Page 406
18.5 Rounding error and integration......Page 410
19.1 Exponential sums with a difference......Page 419
19.2 A major arc estimate......Page 426
19.3 Exponential sums with a large second derivative......Page 428
20.1 Exponential sums......Page 436
20.2 Mean value theorems......Page 442
20.3 The modular form L-function......Page 448
21.1 Introduction......Page 451
21.2 The order of magnitude in the critical strip......Page 454
21.3 The mean square......Page 456
21.4 Gaps between zeros......Page 460
21.5 The twelfth-power moment......Page 464
22.1 Preparation......Page 465
22.2 Type I double sums......Page 469
22.3 Type II double sums......Page 472
22.4 Prime numbers in a smooth sequence......Page 476
Part VI Related work and further ideas......Page 478
23.1 Integer points close to a curve......Page 480
23.2 The Hardy-Littlewood method......Page 483
23.3 Other Farey arc arguments......Page 484
23.4 Higher dimensions......Page 486
23.5 Exponential sums with monomials......Page 488
24.1 Comments on the method......Page 491
24.2 Subdivision without absolute values......Page 492
References......Page 497
Index......Page 504