Geometry of the gauss map and lattice points in convex domains

Abstract

Let Q be a convex planar domain, with no curvature or regularity assumption on the boundary. Let N e (R) -card{RCl e r\Z 2 }, where Q e denotes the rotation of Q by 9. It is proved that, up to a small logarithmic transgression, N e (R) = \Cl\R 2 + O(R 2/i ), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Q under the Gauss map is also obtained.Introduction. Let Q be a bounded convex planar domain, and let = card{RQnZ 2 }. It was observed by Gauss that N(R) = \Cl\R 2 + D(R), where \D(R)\^R, since the discrepancy D(R) cannot be larger than the number of lattice points that live a distance at most 1 A/2 from the boundary of Q.Here, and throughout the paper, A^B means that there exists a uniform C, such that A «£ CB. Similarly, A~B means that A^B and B^A. For a general domain the estimate |£>(./?)|=s/? cannot be improved, as can be seen by taking £2 to be a square with sides parallel to the axis or, more generally, a polygon with a side of rational slope. However, the purpose of this paper is to show that the remainder term is better for almost every rotation of the domain.If the boundary of Q has everywhere non-vanishing Gaussian curvature, better estimates for the remainder term are possible. It is a classical result of Hlawka and Herz that, in that case, \D(R) \^R 2/i , and an example due to Jarnik shows that, without further assumptions, this result is best possible. See, for example, [L] and [J]. If the boundary is assumed to have a certain degree of smoothness, further improvements have been obtained, culminating (at the moment) in a result due to Huxley, see [Hu], which says that, if the boundary of Q. is five times differentiable and has curvature bounded below by a fixed constant, then \D(R)\^R 46/n . This is also the best current result for the circle problem, for which the well-known conjecture is that \D(R)\^R 1/2 + E , and indeed it was observed by Hardy that one cannot do better than R 1/2 times an appropriate power of the logarithm.We have noted that, in general, the trivial estimate |Z)(l?)|=s.R cannot be improved without curvature assumption on the boundary of the domain. For example, it was proved by Randol in [R] that, if Q is given by the equation x'" + x'"^l with m>2, then \D(R)\^R Cm~l)/m , and (m-\)/m cannot be replaced by any smaller number. On the other hand, Colin de Verdiere showed in [C] that, if the boundary of Q. has finite order of contact with its tangent lines, then, for almost every rotation of Q, the corresponding error term \D(R, 6)\^R 2/3 . This result was extended to a certain class of domains, where

Sharp rate of average decay of the Fourier transform of a bounded set

“…where ρB denotes the rotation of B by ρ ∈ S d−1 viewed as an element of SO(d). See, for example, [Ios2001], [BCIPT02], and [BIT01] for a detailed discussion of applications of average decay of the Fourier transform to lattice point problems. Also note that Theorem 1.2 below shows that convexity may be replaced by a C d , d > 1, is greater than d+1 2 , then ∆ B (ρB) has positive Lebesgue measure for almost every ρ ∈ S d−1 viewed as an element of SO(d).…”

Two problems associated with convex finite type domains

“…the proof of Lemma 2.2 below). We remark that it is well known that for almost all rotations A ∈ SO(d) the error terms E AΩ (t) improve, see [5], [31], [32], [23], [15], and [2].…”

“…We may assume that U is the unit cube, and that the support of χ has small diameter. We decompose 2]) so that ψ is equal to 1 on the support of ψ. Denote by T k the convolution operator with Fourier multiplier m δ,k and by R k the convolution operator with Fourier multiplier ψ(δ −1/2…”

Lattice points in large convex planar domains of finite type

“…θ, with δ(p 0 ) > 1/p 0 for a given p 0 > 1, for certain class of convex planar domains whose curvature is allowed to vanish to infinite order. His result was extended, in the paper [1] by Brandolini, Colzani, Iosevich, Podkorytov, and Travaglini, to arbitrary convex planar domains with no curvature or regularity assumption on the boundary.…”