Metric Diophantine approximation with two restricted variables I. Two square-free integers, or integers in arithmetic progressions
1988 ◽
Vol 103
(2)
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pp. 197-206
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Keyword(s):
In this paper, together with [7] and [8], we shall be concerned with estimating the number of solutions of the inequalityfor almost all α (in the sense of Lebesgue measure on Iℝ), where , and both m and n are restricted to sets of number-theoretic interest. Our aim is to prove results analogous to the following theorem (an improvement given in [2] of an earlier result of Khintchine [10]) and its quantitative developments (for example, see [11, 12,6]):Let ψ(n) be a non-increasing positive function of a positive integer variable n. Then the inequality (1·1) has infinitely many, or only finitely many, solutions in integers to, n(n > 0) for almost all real α, according to whether the sumdiverges, or converges, respectively.
1986 ◽
Vol 99
(3)
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pp. 385-394
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2014 ◽
Vol 91
(1)
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pp. 34-40
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Keyword(s):
1960 ◽
Vol 12
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pp. 619-631
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Keyword(s):
1995 ◽
Vol 118
(1)
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pp. 1-5
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2010 ◽
Vol 81
(2)
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pp. 177-185
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Keyword(s):
1939 ◽
Vol 35
(4)
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pp. 527-547
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Keyword(s):
1950 ◽
Vol 46
(2)
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pp. 209-218
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2013 ◽
Vol 94
(1)
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pp. 50-105
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2016 ◽
Vol 160
(3)
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pp. 477-494
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Keyword(s):
1989 ◽
Vol 105
(3)
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pp. 547-558
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