Diophantine Approximations and Directional Discrepancy of Rotated Lattices

Optimization for Lattices and Diophantine Approximations

The L2 discrepancy of irrational lattices

It is well known that, when α has bounded partial quotients, the lattices {( k/N,{kα} )}N−1 k=0 have optimal extreme discrepancy. The situation with the L 2 discrepancy, however, is more delicate. In 1956 Davenport established that a symmetrized version of this lattice has L2 discrepancy of the order √ logN, which is the lowest possible due to the celebrated result of Roth. However, it remained...

Directional Discrepancy in Two Dimensions

In the present paper, we study the geometric discrepancy with respect to families of rotated rectangles. The well-known extremal cases are the axis-parallel rectangles (logarithmic discrepancy) and rectangles rotated in all possible directions (polynomial discrepancy). We study several intermediate situations: lacunary sequences of directions, lacunary sets of finite order, and sets with small ...

On a Generalization of Littlewood’s Conjecture

We present a class of lattices in R (d ≥ 2) which we call GL− lattices and conjecture that any lattice is such. This conjecture is referred to as GLC. Littlewood’s conjecture amounts to saying that Z is GL. We then prove existence of GL lattices by first establishing a dimension bound for the set of possible exceptions. Existence of vectors (GL − vectors) in R with special Diophantine propertie...

Integer Optimization and Lattices

• Lattices. We will see basic properties of lattices, followed by Minkowski’s Theorem which guarantees that any symmetric convex set with volume bigger than 2 must have an non-zero integer point. We will show an application of Minkowski’s theorem to Diophantine approximations. Then we will introduce the powerful concept of Lattice basis reduction which finds an almost orthogonal basis for a lat...