Primitive lattice points in starlike planar sets

Primitive Lattice Points in Starlike

This article is concerned with the number B D (x) of integer points with relative prime coordinates in p x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for B D (x). Applications to certain special geometric and arithmetic problems are discussed .

Primitive sets in a lattice

We study the problem of extending a primitive set of size k < n, in a lattice of rank n, to a basis of the lattice. Our approach relies on solving the matrix completion problem, and we show that the relevant computed matrices have nicely bounded entries. We present formulas, for the cases k = 1, n − 1 for an extended basis, and analyze the computational complexity for each of these two cases. T...

Counting Lattice Points in Admissible Adelic Sets

On the Distribution of Primitive Lattice Points in the Plane

This is one form of Kronecker’s theorem [4-j aura, since N can be chosen arbitrarily large, it follows that there are an infinity of integer sets (x, yl, -. . , y,) with x > 0 satisfying (A). For n > 2, it is not possible to strengthen this result by replacing the & in (A), throughout, by any function y/(x) which tends to zero as x-m (see, e.g., 15) , Kap VII,37, Satz 6). But, in the case n = 1...

Inequalities for Lattice Constrained Planar Convex Sets

Every convex set in the plane gives rise to geometric functionals such as the area, perimeter, diameter, width, inradius and circumradius. In this paper, we prove new inequalities involving these geometric functionals for planar convex sets containing zero or one interior lattice point. We also conjecture two results concerning sets containing one interior lattice point. Finally, we summarize k...