FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 5, PAGES 93-101

On inhomogeneous Diophantine approximation and Hausdorff dimension

M. Laurent

Abstract

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Let $$G =ZA+Zⁿ Ì Rⁿ be a dense subgroup of rank $n+1$ and let $\$\; \backslash hat\; \{\backslash omega\; \}(A)\; \$$ denote the exponent of uniform simultaneous rational approximation to the generating point $A$. For any real number $\$\; v\backslash geq\; \backslash hat\; \{\backslash omega\; \}(A)\; \$$, the Hausdorff dimension of the set $B$_v of points in $$Rⁿ that are $v$-approximable with respect to $$G is shown to be equal to $1/v$.

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Last modified: May 30, 2011