Diophantine property in the group of affine transformations of the line

TL;DR

This paper studies the Diophantine property in the group of affine transformations of the line, showing that non-Diophantine pairs are extremely rare within a specific family, having Hausdorff dimension zero.

Contribution

It establishes that within a certain one-parameter family, the set of non-Diophantine pairs has Hausdorff dimension zero, highlighting their scarcity.

Findings

Non-Diophantine pairs form a set of Hausdorff dimension zero.

The Diophantine property is characterized for pairs in the affine group.

Most pairs in the family satisfy the Diophantine condition.

Abstract

We investigate the Diophantine property of a pair of elements in the group of affine transformations of the line. We say that a pair of elements g_1,g_2 in this group is Diophantine if there is a number A such that a product of length l of elements of the set {g_1,g_2,g_1^{-1},g_2^{-1}} is either the unit element or of distance at least A^{-l} from the unit element. We prove that the set of non-Diophantine pairs in a certain one parameter family is of Hausdorff dimension 0.