Permutations Destroying Arithmetic Structure

Abstract

Given a linear form $C_1X_1 + \cdots + C_nX_n$, with coefficients in the integers, we characterize exactly the countably infinite abelian groups $G$ for which there exists a permutation $f$ that maps all solutions $(\alpha_1, \ldots , \alpha_n) \in G^n$ (with the $\alpha_i$ not all equal) to the equation $C_1X_1 + \cdots + C_nX_n = 0 $ to non-solutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arithmetic progressions. We also study the finite version of the problem suggested by Hegarty. We show that the number of permutations of $\mathbb{Z}/p\mathbb{Z}$ that map all 4-term arithmetic progressions to non-progressions, is asymptotically $e^{-1}p!$.