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Introduction; Part I Elementary Methods; 1 The rational line; 2 Polygons and area; 3 Integer points close to a curve; 4 Rational points close to a curve; Part II The Bombieri-Iwaniec Method; 5 Analytic methods; 6 C Mean value theorems ; 7 The simple exponential sum; 8 Exponential sums with a difference; 9 Exponential sums with a difference; 10 Exponential sums with modular form coefficients; Part III The First Spacing Problem: Integer Vectors; 11 The ruled surface method; 12 The Hardy Littlewood method; 13 The first spacing problem for the double sum; Part IV The Second Spacing Problem: Rational vectors; 14 The first and second conditions; 15 Consecutive minor arcs; 16 C The third and fourth conditions ; Part V Results and Applications; 17 Exponential sum theorems; 18 Lattice points and area; 19 Further results; 20 Sums with modular form coefficients; m 21 Applications to the Riemann zeta function; 22 An application to number theory: prime integer points; Part IV Related Work and Further Ideas; 23 Related work; 24 Further ideas; References

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