Result: Solution-free sets for linear equations: Solution-free sets for linear equations.
Further Information
Let \(G\) be an Abelian group. If \(A\) is a subset of \(G\), and \(k\)-positive integer, \(k> 1\), then \(A\) is called strongly \(k\)-sum free if \(A\) does not contain the sum of \(r\) its elements for \(1< r< k+1\). In the first part the authors study the cardinality of strongly \(k\)-sum free subsets of a finite Abelian group. In the second part the authors use these results for the remainder class modulo \(m\), \(m\) a positive integer and derive some results about strongly \(k\)-sum free subsets of the set of positive integers. Theorem 9 deals with the upper asymptotic density of strongly \(k\)-sum free sets of positive integers. The object of observation in the last part is the set of positive integers containing no solutions to the equation \(x= y+ az\), \(a\) is given. Several results are improvement of results from \textit{T. Łuczak} and \textit{T. Schoen} [ibid. 66, 211--224 (1997; Zbl 0884.11018)].