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{\bf Tomasz Schoen}
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{\bf A Note on the Number of $(k,l)$-Sum-Free Sets}
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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 $\varphi(x)=\varphi_x(x)$.


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