Higher dimensional Steinhaus and Slater problems via homogeneous dynamics

Abstract

The three gap theorem, also known as the Steinhaus conjecture or three distance theorem, states that the gaps in the fractional parts of α, 2α, . . . , N α take at most three distinct values. Motivated by a question of Erdős, Geelen and Simpson, we explore a higher-dimensional variant, which asks for the number of gaps between the fractional parts of a linear form. Using the ergodic properties of the diagonal action on the space of lattices, we prove that for almost all parameter values the number of distinct gaps in the higher dimensional problem is unbounded. Our results in particular improve earlier work by Boshernitzan, Dyson and Bleher et al. We furthermore discuss a close link with the Littlewood conjecture in multiplicative Diophantine approximation. Finally, we also demonstrate how our methods can be adapted to obtain similar results for gaps between return times of translations to shrinking regions on higher dimensional tori.

Higher dimensional gap theorems for the maximum metric

“…To prove Theorem 1 we will first realize the quantity g * N as the value of a function G defined on the space SL(d + 1, Z)\ SL(d + 1, R) of unimodular lattices in R d+1 . This part of the proof, carried out in Section 2, is exactly analogous to the development in [2] and [3], which in turn is an extension of ideas originally presented by Marklof and Strömbergsson in [4]. In Section 3 we will use a simple geometric argument to bound G(M ), when M is an arbitrary unimodular lattice in R d , and for d = 2 and 3 we will give examples of L, α, and N for which our upper bounds are attained.…”

A three gap theorem for the adeles

“…The classical three gap theorem (also known as the three distance theorem and as the Steinhaus problem) asserts that, for any α ∈ R and N ∈ N, the collection of points nα mod 1, 1 ≤ n ≤ N, partitions R/Z into component arcs having one of at most three distinct lengths. This theorem was first proved independently in the 1950's by Sós [8,9], Surányi [10], and Świerczkowski [11], and it has since been reproved numerous times and generalized in many ways (see the introductions and bibliographies of [2,3]).…”

“…Our proof of this theorem is an adaptation to the adeles of the lattice based approach to gaps problems in Diophantine approximation first introduced by Marklof and Strömbergsson in [6] to give a new proof of the three gap theorem. The utility of their approach lies in its flexibility for generalization to higher dimensional problems where other techniques do not work well (see [2,3,5]).…”

Representation and coding of rational pairs on a Triangular tree and Diophantine approximation in $\mathbb{R}^2$

“…We believe that the Triangular tree will be useful for the study of interesting phenomena related to two-parameter systems, as this is the case for the Farey tree and one-parameter systems. For recent investigations on higher-dimensional phenomena we refer to [3,4,10].…”