EXPONENTIAL SUMS WITH A PARAMETER M. N. HUXLEY and N. WATT [Received 12 August 1988—Revised 6 December 1988] ABSTRACT Let F(x,y) be a real function with sufficiently many derivatives existing and satisfying certain non-vanishing conditions for 1 ^x **2, O^y « 1. Let y run through a discrete set of values y ( . The exponential sum (over a subset of [M, 2M) which may depend on i) is smaller on average than the best estimate in [9] for the individual sums. Thus in a double exponential sum we obtain a small saving in the second summation. This leads to better estimates for short sums in one variable. We use the 'discrete Hardy-Littlewood method' of Bombieri and Iwaniec as developed in [4,9]. 1. Introduction We develop a theme from our first paper [4] on the estimation of the exponential sum 5= I e( TF Q) (1.1, m=M \ \M// as M, M 2 , T tend to infinity with M 2 < 2M < T. Sums of this type appear whenever Fourier analysis is used to compare the actual number of sets of integers satisfying some non-linear condition with the expected number. Perhaps the most interesting example is Pyatetski-Shapiro's result that there are infinitely many primes in the sequence of integer parts of the numbers n 1+6 as n runs through the integers, when 8 is positive but sufficiently small. These sums also arise in the theoretical discussion of rounding error in numerical integration. A special case of this is the classical problem of the number of lattice points (integer points) in a large circle, discussed in [3] and [5]. In both these cases the sum to be considered is a multiple sum over two or more integer variables. Graham and Kolesnik [2] are preparing a full account of the exponential sums method. In [4] we used Bombieri and Iwaniec's method [1], which echoes the Hardy-Littlewood method for integrals of exponential sums, and is very much a one-variable method. However in § 6 of [4] we considered a function F(x, y) of two variables with 1=^x^2, O^y^l, evaluated at a discrete set of values y\> ••-, yL with yi+\ — yi ^1/U. The corresponding exponential sums are M 2 (i) Si= 2 e{f{m, yi )), (1.2) w=A/,(i) where M =£ M x (i) ^ M 2 (i) < 2M and f(x, y) = TF(x/M,y). The results of this paper can be be summed up as follows: if F(x, y) is sufficiently differentiate, and certain polynomials in the partial derivatives do not vanish, then |5,| is smaller on average than our best estimate for the individual values. In particular if y t = IIL, for / = 1,..., L, Theorem 2 gives an estimate for a double exponential sum, with A.M.S. (1980) subject classification: primary 11L40; secondary 11M06, 11P55. Proc. London Math. Soc. (3) 59 (1989) 233-252.