Linear equations and sets of integers

We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then |A|\leqq 2^-ck/(logk)2N|A|\leqq 2^{-ck/{(\log k)}^{2}}N, for some absolute constant c>0, provided that N is large enough.