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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Dyadic methods in the measure theory of numbers


Author: R. C. Baker
Journal: Trans. Amer. Math. Soc. 221 (1976), 419-432
MSC: Primary 10K15; Secondary 10K05
MathSciNet review: 0417097
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Abstract: Some new theorems in metric diophantine approximation are obtained by dyadic methods. We show for example that if $ {m_1},{m_2}, \ldots $, are distinct integers with $ {m_n} = O({n^p})$ then $ {\Sigma _{n \leqslant N}}e({m_n}x) = O({N^{1 - q}})$ except for a set of x of Hausdorff dimension at most $ (p + 4q - 1)/(p + 2q)$; and that for any sequence of intervals $ {I_1},{I_2}, \ldots $ in [0, 1) the number of solutions of $ \{ {x^n}\} \in {I_n}\;(n \leqslant N)$ is a.e. asymptotic to $ {\Sigma _{n \leqslant N}}\vert{I_n}\vert(x > 1)$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1976-0417097-6
PII: S 0002-9947(1976)0417097-6
Keywords: Dyadic representation of integers, discrepancy modulo one, Hausdorff dimension, strong uniform distribution, fractional parts of sequences
Article copyright: © Copyright 1976 American Mathematical Society