Translation invariant equations and the method of Sanders (Q2918796)

The author considers a linear equation NEWLINE\[NEWLINEc_1 x_1+\cdots+ c_s x_s= 0,\quad c_j\neq 0,\quad s\geq 3\quad\text{and}\quad c_1+\cdots+ c_s= 0.\tag{1}NEWLINE\]NEWLINE Extending the result of Sanders. The author proves:NEWLINENEWLINE Theorem. The sine of the largest subset of \(A\in\{1,\dots, N\}\) having no solutions of equation (1) differ from the form \((a_1,\dots, a)\in A^s\) is NEWLINE\[NEWLINE|A|= O\left(N\left({(\log\log N)^s\over\log N}\right)^{s-2}\right).NEWLINE\]NEWLINE A similar result is obtained in the ring of polynomials \(\mathbb F_q[t]\) over a finite field.