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 $\gamma_1,\gamma_2$ in this group is Diophantine if there is a number $A$ such that a product of length $l$ of elements of the set $\{\gamma_1,\gamma_2,\gamma_1^-1,\gamma_2^-1\}$ 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~$0$.

DOI: 10.14232/actasm-013-757-6

AMS Subject Classification (1991): 22E25, 30C15, 11C08

Keyword(s): solvable Lie groups, Diophantine property, roots of polynomials, zeros of polynomials

Received January 24, 2013, and in revised form March 3, 2013. (Registered under 7/2013.)