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ON ADDITIVE COMPLEMENTS IN THE COMPLEMENT OF A SET OF NATURAL NUMBERS
Published online by Cambridge University Press: 07 November 2025
Abstract
Let A be a set of natural numbers. A set B of natural numbers is an additive complement of the set A if all sufficiently large natural numbers can be represented in the form 
$x+y$, where 
$x\in A$ and 
$y\in B$. We establish that if 
$A=\{a_i: i\in \mathbb {N}\}$ is a set of natural numbers such that 
$a_i<a_{i+1} $ for 
$i \in \mathbb {N}$ and 
$\liminf _{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set 
$B\subset \mathbb {N}$ such that 
$B\cap A = \varnothing $ and B is a sparse additive complement of the set A.
MSC classification
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- Research Article
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc
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