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
