A Note on Inhomogeneous Diophantine Approximation

Abstract

Abstract. Let α be an irrational number. We determine the Hausdorff dimension of sets of real numbers which are close to infinitely many elements of the sequence ({nα}) n ≥ 1 .

A Note on Inhomogeneous Diophantine Approximation With a General Error Function

“…Using the similar idea as that in Bugeaud [2], Schmeling and Troubetzsky [10], we can get the following result (the details are omitted). …”

“…In 2003, Bugeaud [2], Schmeling and Troubetzsky [10] improved, independently, the above result due to Bernik and Dodson as follows.…”

On simultaneous inhomogeneous Diophantine approximation

“…In the "doubly metric" case and in the first "singly metric" case mentioned above, satisfactory answers have been given by Dodson [7] and Levesley [13], respectively. However, no multidimensional extension of the statements established in [14] and [4] has been studied up to now, and it is the purpose of the present work to report various results on this question.…”

“…, α k linearly independent over the rationals and such that the Hausdorff dimension of the set V k (α) is equal to 1/k. In view of the results from [14] and [4], the dimension cannot be smaller. Furthermore, we prove that, for any arbitrarily small positive w, there exist real k-tuples α with 1, α 1 , .…”

“…different approach), by Bugeaud [4] who showed that, for any τ > 1 and any fixed irrational number α, the set T τ (α) := {ξ ∈ R : nα − ξ < 1/n τ for infinitely many n ∈ Z ≥1 } has Hausdorff dimension 1/τ . We stress that this value does not depend on α.…”