Mixing and linear equations over groups in positive characteristic
Let \(G\) be a multiplicative abelian group. The author calls a subset \(\Sigma\) of \(G^n\) a broad subset if (i) \(\Sigma\) is infinite, (ii) for each \(g\) in \(G\) and for each \(i=1,\ldots,n\), there are at most finitely many \((x_1,\ldots,x_n)\) in \(\Sigma\) with \(x_i=g\), and (iii) (for \(n\geq 2\)) for each \(g\) in \(G\) and each \(i\) and \(j\) satisfying \(1\leq i<j\leq n\), there are at most finitely many \((x_1,\ldots,x_n)\) in \(\Sigma\) with \(x_i/x_j=g\). The main result is the following theorem: Let \(K\) be a field of positive characteristic and let \(G\) be a finitely generated subgroup of the set of nonzero elements of \(K\). Suppose that there are \(a_1,\ldots,a_n\) in \(K\) such that the equation \(a_1X_1+\ldots+a_nX_n=1\) has a broad set of solutions in \(G^n\). Then there are \(b_1,\ldots,b_n\) in \(K\) and \(g_1,\ldots,g_n\) in \(G\) with \(g_i\not=1\) for all \(1\leq i\leq n\) and (if \(n\geq 2\)) \(g_i/g_j\not=1\) for all \(1\leq i<j\leq n\), such that the equation \(b_1g_1^k +\ldots+b_ng_n^k=1\) has infinitely many solutions in positive integers \(k\). As a consequence, the author obtains the following corollary: For any positive integer \(d\) let \(\alpha\) be an algebraic \({\mathbb Z}^d\)-action on a compact abelian group and let \(r\geq 2\) be an integer. If every subset \(S\) in \({\mathbb Z}^d\) of cardinality \(r\) is \(\alpha\)-mixing, then \(\alpha\) is \(r\)-mixing.
scientific article; zbMATH DE number 2108118
scientific article