Roth's theorem on systems of linear forms in function fields

Abstract

1. Introduction. For r, s ∈ N = {1, 2, . . .} with s ≥ 2r + 1, let (bi,j ) be an r×s matrix whose elements are integers. Suppose that bi,1+· · ·+bi,s = 0 (1 ≤ i ≤ r). Suppose further that among the columns of the matrix, there exist r linearly independent columns such that, if any of the r columns are removed, the remaining n − 1 columns of the matrix can be divided into two sets so that among the columns of each set there are r linearly independent columns. For k ∈ N, denote by D([1, k]) the maximal cardinality of an integer set A ⊆ [1, k] such that the equations bi,1x1 + · · · + bi,sxs = 0 (1 ≤ i ≤ r) are never satisfied simultaneously by distinct elements x1, . . . , xs ∈ A. Using techniques similar to his work on sets free of three-term arithmetic progressions (see [4]), Roth [5] showed that D([1, k]) k/(log log k) 1/r2 . In this paper, we will build upon the methods in [2] to study an analogous question in function fields