A generalization of Roth's theorem in function fields
The authors obtain a generalization of the well-known Meshulom's result on sets having no solutions of an affine equation in \(\mathbb F_q[t]\). Let \(S_N\) be the subset of \(\mathbb F_q[t]\) of polynomials of degree \(\leq N\). Let \(A\) be a subset of \(S_N\) having no solutions of the equation \(g_1 x_1+\cdots+ g_s x_s= 0\), \(s\geq 3\), \(x_j\in A\), \(j= 1,\dots, s\) are distinct elements. The main result is the following: Theorem. For any \(g_j\in\mathbb F_q[t]\), \(j= 1,\dots, s\), \(s\geq 3\) such that \(g_1+\cdots+ g_s=0\) there is \(C(g_1,\dots, g_s, q)> 0\) with \[ |A|\leq C\,|S_N|\Biggl({(\log\log|S_N|^2\over\log|S_N|}\Biggr)^{{2(s-2)^2\over 4s-9}}, \] where \(A\) contains no solution of the equation \(g_1 x_1+\cdots+ g_s x_s= 0\) with distinct \(x_j\in A\).
sets containing no solutons of linear equations
scientific article; zbMATH DE number 6126104
scientific article