Permutations destroying arithmetic structure

Veselin Jungi´c, Julian Sahasrabudhe
Given a linear form C 1 X 1 + · · · + C n X 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 (α 1 ,. .. , α n) ∈ G n (with the α i not all equal) to the equation C 1 X 1 + · · · + C n X 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
more » ... blem suggested by Hegarty. We show that the number of permutations of Z/pZ that map all 4-term arithmetic progressions to non-progressions, is asymptotically e −1 p!.