Vol. 85, No. 2, 1979

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On the completeness of sequences of perturbed polynomial values

Stefan Andrus Burr

Vol. 85 (1979), No. 2, 355–360

If S is an arbitrary sequence of positive integers, define P(S) to be the set of all integers which are representable as a sum of distinct terms of S. Call a sequence S complete if P(S) contains all sufficiently large integers, and subcomplete if P(S) contains an infinite arithmetic progression. We will prove the following theorem: Let n-th term of the integer sequence S have the form f(n) + O(nα), where f is a polynomial and where 0 α < 1; then S is subcomplete. We further show that S is complete if, in addition, for every prime p there are infinitely many terms of S not divisible by p. (We call any sequence satisfying this last property an R-sequence.) We will then extend these results to considerably more general sequences.

Mathematical Subject Classification
Primary: 10L05, 10L05
Received: 12 July 1977
Revised: 11 May 1979
Published: 1 December 1979
Stefan Andrus Burr