Abstract: We investigate arithmetic progressions in sparse sets. We further develop the theory of arithmetic progressions in M-lacunary sequences. In particular, there is a finite constant K , where N N 1 3 , such that if A is a finite M-lacunary sequence and the sum of the reciprocals of the elements of A is greater than or equal to K , then A contains N consecutive terms in arithmetic N progression. An investigation of arithmetic progressions in geometric progressions {f (n) 1 = {cn) , where c > 1 is a real number and n is a non-negative integer is macTe. The topological structure of the set of all such c , where {cn) contains a 3-term arithmetic progression is discussed. Arithmetic progressions in quadratics {f (n) = {an2 + Bn + y}, where a > 0 are also studied. 2 In particular, necessary and sufficient conditions for ((an + b) ) to contain a three term arithmetic progression are given for most cases of a and b , where a > b 1 0 and (arb) = 1. Some unsolved problems are mentioned.
Publication Year: 1986
Publication Date: 1986-01-01
Language: en
Type: dissertation
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