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On the fractional parts of a set of points
Author(s) -
Cook R. J.
Publication year - 1972
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300004940
Subject(s) - citation , mathematics , set (abstract data type) , combinatorics , information retrieval , computer science , library science , programming language
Heilbronn [5] proved that for any e > 0 there exists C (e) such that for any real θ and N ≥ 1 there is an integer x satisfying where ‖α‖ denotes the difference between α and the nearest integer, taken positively. The result is uniform in θ and so analogous to Dirichlet's inequality for the fractional parts of n θ. The result has been generalized to simultaneous approximations by Danicic [1] and Ming-chit Liu [6]. Here we shall extend the result to any finite number of simultaneous approximations when x 2 is replaced by a k -th power.