کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
6415325 1630652 2016 12 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Variations on Dirichlet's theorem
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
Variations on Dirichlet's theorem
چکیده انگلیسی

We give a necessary and sufficient condition for the following property of an integer d∈N and a pair (a,A)∈R2: There exist κ>0 and Q0∈N such that for all x∈Rd and Q≥Q0, there exists p/q∈Qd such that 1≤q≤Q and ‖x−p/q‖≤κq−aQ−A. This generalizes Dirichlet's theorem, which states that this property holds (with κ=Q0=1) when a=1 and A=1/d. We also analyze the set of exceptions in those cases where the statement does not hold, showing that they form a comeager set. This is also true if Rd is replaced by an appropriate “Diophantine space”, such as a nonsingular rational quadratic hypersurface which contains rational points. Finally, in the case d=1 we describe the set of exceptions in terms of classical Diophantine conditions.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Number Theory - Volume 162, May 2016, Pages 11-22
نویسندگان
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