Result: On sets of natural numbers without solution to a noninvariant linear equation

Let \(a_1, \dots{}, a_k, b\) be integers such that \(a_1+\dots{}+a_k\neq 0\) or \(b\neq 0\). Let \(r(n)\) denote the maximal cardinality of a set of integers in \([1,n]\) without any solution to the equation \(a_1x_1+\dots{}+a_kx_k=b\) in \(x_i\in A\), and let \(\overline \lambda = \limsup r(n)/n\). Further let \(\overline \Lambda = \sup \overline d(A)\) over sets of positive integers without a solution, and define \(\underline \lambda , \underline \Lambda \) analogously with \( \liminf \) and \( \underline d\). All four quantities are positive. The author constructs equations where these quantities behave rather differently. In particular, he gives examples where \(\overline \Lambda 1-\varepsilon \), an equation with \(s\neq 0\) and \(\overline \lambda