Linear equations and sets of integers (Q1928065)

This paper deals with linear equations over integers. In the first result (Theorem~1), the author proves a conjecture of \textit{I. Z. Ruzsa} [Acta Arith. 72, No. 4, 385--397 (1995; Zbl 1044.11617)]: for a noninvariant equation \(R(N)=r(N)+o(N)\). The second result (Theorem~2 -- also related to the previous mentioned paper) shows that for every \(k\geq 2\) there exists a noninvariant equation in \(k\) variables such that \(\lambda =\limsup\frac{r(N)}{N}<2^{-ck/(\log k)^{2}}\) for some absolute constant \(c>0\).