The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri–Iwaniec–Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri–Iwaniec–Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton‐Dyer's approximation determinant method. In the discrepancy estimate O(RK (log R)Λ) in terms of the maximum radius of curvature R, we reduce K from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes K/2 =131/416. 2000 Mathematics Subject Classification 11P21, 11L07.

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The area A inside a simple closed curve C can be estimated graphically by drawing a square lattice of sides 1/M. The number of lattice points inside C is approximately AM 2 . If C has continuous non-zero radius of curvature, the number of lattice points is accurate to order of magnitude at most M a for any oc > §. We show that if the radius of curvature of C is continuously differentiate, then the exponent § may be replaced by j \ , extending the result of Iwaniec and Mozzochi [4] in which C was a circle. On the way we obtain results on two-dimensional exponential sums, the average rounding error of the values of a smooth function at equally spaced arguments, and the number of lattice points close to a smooth arc.

A Van der Corput exponential sum is S = Σ exp (2 π i f(m)) where m has size M, the function f(x) has size T and α = (log M) / log T < 1. There are different bounds for S in different ranges for α. In the middle range where α is near 1/over 2, S=Ofalse(MTθ+ϵfalse). This θ bounds the exponent of growth of the Riemann zeta function on its critical line Re s = 1/over 2. Van der Corput used an iteration which changed α at each step. The Bombieri–Iwaniec method, whilst still based on mean squares, introduces number‐theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of y = f′(x) coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1–40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ‘resonance curve’, one of a family of curves in some dual space, now calculated accurately in the paper ‘Resonance curves in the Bombieri–Iwaniec method’, which is to appear in Funct. Approx. Comment. Math. We turn the whole Bombieri–Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for S. Ends and cusps of resonance curves are treated separately. Bounds for sums of type S lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton‐Dyer's method is stronger. We improve θ from 0.156140… in the previous paper and monograph to 0.156098…. In fact (32/205 + ε, 269/410 + ε) is an exponent pair for every ε > 0. 2000 Mathematics Subject Classification 11L07 (primary), 11M06, 11P21, 11J54 (secondary).

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