An algorithm for complements of finite sets of integers

Author:
Gerald Weinstein

Journal:
Proc. Amer. Math. Soc. **55** (1976), 1-5

MSC:
Primary 10L05

DOI:
https://doi.org/10.1090/S0002-9939-1976-0435023-6

MathSciNet review:
0435023

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Abstract: Let and be sets of nonnegative integers of elements and infinitely many elements, respectively. Suppose has asymptotic density . If, for every integer , we can find such that , then we say that has a complement of density .

Given and there is no known algorithm for determining if such a set exists.

We define regular complement and give an algorithm for determining if exists when complement is replaced by regular complement. More precisely, given and we give an algorithm for determining if has a regular complement with density . We relate this result to the Conjecture. Every has a complement of density .

**[1]**D. J. Newman,*Complements of finite sets of integers*, Michigan Math. J.**14**(1967), 481–486. MR**0218324****[2]**G. Weinstein,*Some covering and packing results in number theory*, J. Number Theory**8**(1976), no. 2, 193–205. MR**0435022**, https://doi.org/10.1016/0022-314X(76)90101-3

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DOI:
https://doi.org/10.1090/S0002-9939-1976-0435023-6

Article copyright:
© Copyright 1976
American Mathematical Society