Vol. 85, No. 2, 1979

Recent Issues
Vol. 332: 1
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
On the completeness of sequences of perturbed polynomial values

Stefan Andrus Burr

Vol. 85 (1979), No. 2, 355–360
Abstract

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
Milestones
Received: 12 July 1977
Revised: 11 May 1979
Published: 1 December 1979
Authors
Stefan Andrus Burr