Trigonometric polynomials and lattice points

Source abstract

In this paper we study the distribution of lattice points on arcs of circles centered at the origin. We show that on such a circle of radius R R , an arc whose length is smaller than 2 R 1 / 2 − 1 ( 4 [ m / 2 ] + 2 ) \sqrt 2 {R^{1/2 - 1(4[m/2] + 2)}} contains, at most, m m lattice points. We use the same method to obtain sharp L 4 {L^4} -estimates for uncompleted, Gaussian sums