Permutations Destroying Arithmetic Structure

Permutations Destroying Arithmetic Structure

Given a linear form C1X1 + · · · + CnXn, 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 (α1, . . . , αn) ∈ Gn (with the αi not all equal) to the equation C1X1+ · · ·+CnXn = 0 to non-solutions. This generalises a result of Hegarty about permutations of an abelian group avoiding arit...

Permutations Destroying Arithmetic Progressions in Finite Cyclic Groups

A permutation π 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 (π(a), π(b), π(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Zn, for all n 6∈ {2, 3, 5, 7}. Here we prove, as a special case of a more gene...

Quasirandom Arithmetic Permutations

In [9], the author introduced quasirandom permutations, permutations of Zn which map intervals to sets with low discrepancy. Here we show that several natural number-theoretic permutations are quasirandom, some very strongly so. Quasirandomness is established via discrete Fourier analysis and the ErdősTurán inequality, as well as by other means. We apply our results on Sós permutations to make ...

Permutations Avoiding Arithmetic Patterns

A permutation π of an abelian group G (that is, a bijection from G to itself) will be said to avoid arithmetic progressions if there does not exist any triple (a, b, c) of elements of G, not all equal, such that c − b = b − a and π(c) − π(b) = π(b) − π(a). The basic question is, which abelian groups possess such a permutation? This and problems of a similar nature will be considered. 1 Notation...

Permutations Avoiding Arithmetic Progressions