Area, Lattice Points and Exponential Sums

Transcription

1 Area, Lattice Points and Exponential Sums Martin N. Huxley School of Mathematics, University of Wales, College of Cardiff, Senghenydd Road Cardiff CF2 4AG. Wales, UK Suppose you have a closed curve. How do you find the area inside? While I was writing my first paper on exponential sums and lattice points, my seven year old daughter came home from school and said. "I know how you find the area of a curve. You count the squares". In other words, copy the curve onto squared paper and count how many squares lie inside the curve. If you want the area to greater accuracy, use paper with smaller squares. If the side of the squares is 1/M, and you count N squares, then the area A is approximately N/M 2. As a number theorist I prefer to take the squares as unit squares, the curve as enlarged by a factor M, and the relation as being that N is approximately AM 2. Some squares are inconveniently cut by the curve. When do you count them? Rule 1. Count a square if its centre lies inside. Rule 2. Count a square if its lower left corner lies inside. Rule 3. Count all incomplete squares as half a square. Rules 1 and 2 are really the same rule: if we shift the squared paper by half a unit in the x and y directions, then the corners of the squares are now where the centres were before. Rule 3 is locally like Rule 2, but it counts extra squares where the curve has maxima and minima, and where x takes its extreme values. Take Rule 2 as the basic rule. The corners of squares are the points in two dimensional space with integer coordinates x, y, the lattice points. How accurate is the rule? If the curve fits into a rectangular box C high, B broad (in terms of unit squares), then the number of squares cut is at most 2B + 2C + 4. As M tends to infinity, this discrepancy has order of magnitude M, whilst the area has order M 2. Lattice points on the curve limit the accuracy. Whether by convention one counts them in or out, changing M to M + e makes little change to the area, and counts them all in, whilst changing M to M e counts them all out. If the curve is a polygon, especially a rectangle with sides parallel to the axes, there can be M lattice points on the sides of the rectangle, so that order of magnitude M is then best possible. So take a smooth curve, convex for convenience. What is the mathematics available? The curve is described in a coordinate-free way by an equation connecting the arc length s with the tangent angle \j/, the radius of curvature being ds/di/j. The Proceedings of the International Congress of Mathematicians, Kyoto, Japan, 1990 tf*\ TUP \Aaihpmafinal Cnnipf««f Innati 1 QQ 1

2 414 Martin N. Huxley lattice of integers has an algebraic isomorphism group SL(2, Z), the two by two integer matrices of determinant one. The lattice is also periodic in two dimensions, so that you can take Fourier transforms. These tell you that for a smooth curve the discrepancy N AM 2 has root mean square lying between bounded multiples of y/m (bounded in terms of the shape of the curve). The mean is taken over translations of the curve. As a consequence, for some constant c, \N-AM 2 \ <cjm for at least half the positions of the curve on the squared paper. The two dimensional Fourier transform does not help much with pointwise upper bounds, even for a circle centred on a lattice point, the Gauss circle problem. The upper bound so far corresponds to replacing the curve by a step function. Voronoi and Sierpinski used Archimedes' idea of replacing the curve by a polygon. The polygon is chosen so that the gradients of its sides are small rational numbers. Small is measured by the height norm: the height of a/q in lowest terms is max( fl, \q\). As far as I know, they did not treat the general curve, but only the hyperbola (for the Dirichlet divisor problem) and the circle. If you assume merely that the curve has a radius of curvature bounded away from zero and infinity, then you can show that the discrepancy \N AM 2 \ has order of magnitude at most (M log 2 M) 2/3 by using the estimate for the discrepancy of a right-angled triangle in terms of the continued fraction expansion.of the gradient a/q. Sierpinski had this result in the Gauss circle problem without the logarithmic factor, so being a perfect circle is a little help. The sides of the polygon have rational gradients, so that they may contain many lattice points. The next step is obvious with hindsight: instead of approximating the curve by a piecewise constant or piecewise linear function, approximate by a piecewise quadratic function. Here the argument becomes highly analytic. Define e(t) to be exp 27iit, and g(t) to be the row-of-teeth function, with o(t) = 1/2 t for 0 < t < 1, and with period one. Then g(t) has the Fourier series The number of integers n in an interval a<n<b is b a + g(b) g (a). I can now explain Iwaniec and Mozzochi's attack on the circle problem (1987). It is closely related to Bombieri and Iwaniec's great paper (1986) on the size of the Riemann zeta function. Bombieri and Iwaniec's paper turns out to be related to Jutila's work on the Dirichlet series of modular forms. The connection is that modular forms are functions of two dimensional lattices. The first step is to divide the curve into arcs corresponding to the sides of the Voronoi-Sierpinksi polygon, and to take a new basis for the integer lattice so that (q, a) is a basis vector. This brings in a 2 x 2 integer matrix. The curve is then approximated as a quadratic in a coordinate n in the direction of the vector (q, a) tangential to the curve.

3 Area, Lattice Points and Exponential Sums 415 The sum involving g(t) gives an exponential sum in two summands n and h when the Fourier series for g(t) is truncated. The next two steps are Poisson summation in each variable. There is a serious danger of error terms adding up. It is lucky that the integral to compute is a Bessel function of order one half which has an exact expression of the form e(t)/jt. After approximating, the main terms involve an inner product: where x ik,i) is a four dimensional vector c(x ( *',).j> c/) ), (M, h 1y/K 1/y/k) indexed by two integers k and /, and y U) is a four dimensional vector constructed from the quadratic approximation. The index j refers to the arc of the curve. If a/q has small height (a major arc), then k and / take a bounded number of values, and you estimate trivially (in fact one need not go so deep). If a/q has large height (a minor arc), then k and / run through complicated ranges. When these have been simplified, the next step is a form of the large sieve. The modern large sieve is best described as a sort of Sobolev inequality relating the discrete L 2 norm of a function to the ordinary L 2 norms of the function and its derivatives. The discrete L 2 norm in one dimension is sup /(z,) 2 S S taken over sets S = {z l9..., z R ) with \z { zj\ > ö for / + j. In number-theoretic applications we have a fixed set S, usually corresponding to a subset of the rational numbers, and the function f(z) is an exponential sum, which can be regarded as an integral with respect to a discrete measure supported on the integers. In this case there is a duality principle and an underlying bilinear form, in which the vectors x are on the same footing as the vectors y. A further generalisation replaces the condition \z {, Zj\ > ö by a factor in the upper bound which counts the number of pairs of points differing by less than ö (the convolution of the discrete measure supported on the points of S with itself, against a kernel function supported on a ^-neighbourhood of the diagonal). The next two steps are counting the number of coincidences among sums of pairs of x vectors, which is number theory, and the number of coincidences among pairs of y vectors, which is number theory mixed with analysis. A coincidence between two y vectors means that on two different arcs, the curve weaves between lattice points in the same way. A large set of mutually coincident vectors will add up to a systematic error in the discrepancy N AM 2. Rather than write down the conditions for entries of the y vectors to coincide, I try to describe them. A y vector corresponds to a minor arc and a two by two integer matrix. Coincidence in the first entries means that the two integer matrices

4 416 Martin N. Huxley are close in the sense that FQ -1 is a matrix with small integer entries, and so PQ~ l is a short word in the generators I J and f J. Coincidence in the second entry involves the matrix, the constant term and the gradient. Coincidence in the third entry is a relation between the denominators q of the gradients a/q and the coefficients of n 2 in the quadratic approximations. Coincidence in the fourth entry involves all the coefficients of the approximating polynomial, but if the other three entries coincide, then the condition simplifies to one involving the constant terms and the denominators q of the gradients a/q. What arguments can one use in counting the number of pairs of coincident vectors? The first is compactness, or in its discrete version, Dirichlet's box principle. If there are many points in a bounded region, then there is a small set containing many of them. The second principle is approximation, or the mean value theorem. A smooth function on a small set can be approximated by a polynomial, or still better, a linear function. The third principle is the boon and bane of analytic number theory. The only arbitrarily small integer is zero. An inequality which is strengthened too much turns into an equation. When these three arguments are used in turn, we pass from discrete to continuous and back to discrete again. At present we are overestimating the number of coincidences because we cannot handle the constant term in the quadratic polynomial. So the second and fourth entries of the y vectors are not used. Iwaniec and Mozzochi (1987) could count coincidence only for the rectangular hyperbola. Huxley and Watt (1988) generalised the relevant lemma in Bombieri and Iwaniec (1986), getting bounds for one dimensional exponential sums, and Huxley (1990) adapted the Iwaniec-Mozzochi method to general smooth closed curves to get N = AM 2 + 0(M 7/11 (log M) 47/22 ). Can this be improved? There is some hope of using the second and fourth entries of the y vectors, which would give a small improvement on the exponent 7/11. To reach one half one must avoid taking moduli. Bombieri and Iwaniec (1986) suggest using the Fourier theory of the modular group, which acts on the vectors (q, a) corresponding to the sides of the Voronoi-Sierpinski polygon. This Fourier theory was suggested by Selberg (1956) and worked into a usable form by Kuznietsov (1980) and Deshouillers and Iwaniec (1982). There are many- related questions. Bombieri and Pila (1989) have shown by algebraic geometry that if the curve is not algebraic, then the number of integer points on it is 0(M E ) for any positive e, improving the earlier result of Swinnerton- Dyer (1974). Huxley (1988,1989) has considered the number of integer points within a small distance ö of the curve, using a mixture of elementary and exponential sum techniques. The lattice point problem is a special case of rounding error in numerical approximation, and one can give estimates there too (Huxley 1991). This method seems to be purely two dimensional, because a quadratic approximation to a surface contains too many terms. Even the continued fraction rule does not generalise. But I hope that someone will take up the challenge of finding an arithmetic method for counting lattice points in three or more dimensions.

5 Area, Lattice Points and Exponential Sums 417 References Bombieri, E. (1987): Le grand crible dans le théorie analytique des nombres, 2de edn. Astérisque, Paris Bombieri, E., Iwaniec, H. (1986): On the order of Ç(l/2 + it). Ann. Sc. Norm. Sup. Pisa CI. Sci. (4) 13, Bombieri, E., Pila, J. (1989): The number of integral points on arcs and ovals. Duke Math. J. 59, Deshouillers, J.-M., Iwaniec, H. (1982): Kloosterman sums and Fourier coefficients of cusp forms, Invent, math. 70, Fouvry, E., Iwaniec, H. (1989): Exponential sums for monomials, J. Number Theory 33, Heath-Brown, D.R., Huxley, M.N. (1990): Exponential sums with a difference. Proc. London Math, Soc. (3) 61, Huxley M.N. (1988): The fractional parts of a smooth sequence. Mathematika 35, Huxley M.N. (1989): The integer points close to a curve. Mathematika 36, Huxley M.N. (1990): Exponential sums and lattice points. Proc. London Math. Soc. (3) 60, Huxley M.N. (1991): Exponential sums and rounding error. J. London Math. Soc. (to appear) Huxley, M.N., Kolesnik, G. (1991): Exponential sums and the Riemann zeta function III. Proc. London Math. Soc. (3) 62 (to appear) Huxley, M.N., Watt, N. (1988): Exponential sums and the Riemann zeta function. Proc. London Math. Soc. (3) 57, 1-24 Huxley, M.N., Watt, N. (1989): Exponential sums with a parameter. Proc. London Math. Soc. (3) 59, Iwaniec, H., Mozzochi, C.J. (1988): On the divisor and circle problems. J. Number Theory 29, Jutila, M. (1987): Lectures on a method in the theory of exponential sums. Tata Institute Lectures in Mathematics and Physics, vol. 80. Springer, Bombay Krätzel. E., (1988): Lattice points. D.V.W., Berlin Kuznietsov, N.V. (1980): Peterson's conjecture for cusp forms of weight zero and Linnik's conjecture on sums of Kloosterman sums. Mat. Sbornik Selberg, A. (1956): Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, Sierpinski, W. (1906): Sur un problème du calcul des fonctions asymptotiques. Prace Mat.-Fiz 17, Swinnerton-Dyer, H.P.F. (1974): The number of lattice points on a convex curve. J. Number Theory 6, Voronoi, G. (1906): Sur un problème du calcul des fonctions asymptotiques. J. Reine Angew. Math. 126 Watt, N. (1989a): A problem on semicubical powers. Acta Arith. 52, Watt, N. (1989b): Exponential sums and the Riemann zeta function II, J. London Math. Soc. 39, Watt, N. (1990): A problem on square roots of integers. Periodica Math. Hung. 21, Watt, N. (1991): Exponential sums with a character. (To appear)

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