Permutations destroying arithmetic progressions in finite cyclic groups

TL;DR

This paper proves the existence of permutations in finite cyclic groups that destroy all non-trivial 3-term arithmetic progressions, confirming a long-standing conjecture for sufficiently large n and specific prime cases.

Contribution

It establishes the existence of AP-destroying permutations for all large enough n and certain prime orders, advancing understanding of combinatorial structures in cyclic groups.

Findings

Existence of AP-destroying permutations for all n >= approximately 1.4 x 10^{14}

Constructs such permutations for primes p > 3 with p ≡ 3 (mod 8)

Confirms a conjecture from 2004 for large n and specific prime cases

Abstract

A permutation \pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).