Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {A_k} = \{ 0,{a_2},{a_3}, \ldots ,{a_k}\} $ and $ B = \{ 0,{b_2},{b_3}, \ldots \} $ be sets of nonnegative integers of $ k$ elements and infinitely many elements, respectively. Suppose $ B$ has asymptotic density $ x:d(B) = x$. If, for every integer $ n \geqq 0$, we can find $ {a_i} \in {A_k},{b_j} \in B$ such that $ n = {a_i} + {b_j}$, then we say that $ {A_k}$ has a complement of density $ \leqq x$.

Given $ {A_k}$ and $ x$ there is no known algorithm for determining if such a set $ B$ exists.

We define regular complement and give an algorithm for determining if $ B$ exists when complement is replaced by regular complement. More precisely, given $ {A_4}$ and $ x = 1/3$ we give an algorithm for determining if $ {A_4}$ has a regular complement $ B$ with density $ \leqq 1/3$. We relate this result to the Conjecture. Every $ {A_4}$ has a complement of density $ \leqq 1/3$.


References [Enhancements On Off] (What's this?)

  • [1] D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967), 481-486. MR 36 #1411. MR 0218324 (36:1411)
  • [2] G. Weinstein, Some covering and packing results in number theory, J. Number Theory (to appear). MR 0435022 (55:7984)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 10L05

Retrieve articles in all journals with MSC: 10L05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0435023-6
Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society