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.)
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