Extensions of the Kinchine Wisser Theorem

Cover Extensions of the Kinchine Wisser Theorem
Extensions of the Kinchine Wisser Theorem
Harold N Shapiro
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•i-v). *fl'r nl"e. T 13 Theorem 1;. 2 Given an Infinite sequence such that ^(^ ) > 0, and any infinite sequence of real numbers e, 1 = 1, 2, . , . , 1), and any A e ^ which appears in the sequence after all the A, , ^l = l, c. K, we have (1+. 2) P^ /-, n a/A) > (1- e^)P(A) ^1 "• ^k We note that Theorem y. \ is simply the special case of Theorem \\, 2 in which all the e. = e. The added strength of Theorem \\. 2 lies in the fact that e. May be chosen so as to tend to zero as i — > oo, and as rap...idly as we please* Fe shall in fact obstain the following slightly stronger assertion: Theorem )-t.. 2A . Given an infinite sequence "y!' such that /\(^) > 0, and any function (|)(u) > 0, such that (j)(u) — > as u- — > CO, there exists an infinite subse- quence ii^ .^, ?) such that for all riots a •* *'.
•rol i: &;' - ii y;aflj rf J ^0e5 I ' ■■ . 1^ -f ^r ^ r^sf.
d-i s?'!)'^ O? fTC?.


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