A Note on the Number of $(k,l)$-Sum-Free Sets

Abstract

A set $A\subseteq {\bf N}$ is $(k,\ell)$-sum-free, for $k,\ell\in {\bf N}$, $k>\ell$, if it contains no solutions to the equation $x_1+\dots+x_k=y_1+\dots+y_{\ell}$. Let $\rho=\rho (k-\ell)$ be the smallest natural number not dividing $k-\ell$, and let $r=r_n$, $0\le r < \rho$, be such that $r\equiv n \pmod {\rho }$. The main result of this note says that if $(k-\ell)/\ell$ is small in terms of $\rho$, then the number of $(k,\ell)$-sum-free subsets of $[1,n]$ is equal to $(\varphi(\rho)+\varphi_r(\rho)+o(1)) 2^{\lfloor n/\rho \rfloor}$, where $\varphi_r(x)$ denotes the number of positive integers $m\le r$ relatively prime to $x$ and $\varphi(x)=\varphi_x(x)$.