Words avoiding repetitions in arithmetic progressions

Words avoiding repetitions in arithmetic progressions

Carpi constructed an infinite word over a 4-letter alphabet that avoids squares in all subsequences indexed by arithmetic progressions of odd difference. We show a connection between Carpi’s construction and the paperfolding words. We extend Carpi’s result by constructing uncountably many words that avoid squares in arithmetic progressions of odd difference. We also construct infinite words avo...

Permutations Avoiding Arithmetic Progressions

On permutations avoiding arithmetic progressions

Let S be a subset of the positive integers, and let σ be a permutation of S. We say that σ is a k-avoiding permutation of S if σ does not contain any k-term AP as a subsequence. Similarly, the set S is said to be k-avoidable if there exists a k-avoiding permutation of S.

On Pansiot Words Avoiding 3-Repetitions

The recently confirmed Dejean’s conjecture about the threshold between avoidable and unavoidable powers of words gave rise to interesting and challenging problems on the structure and growth of threshold words. Over any finite alphabet with k ≥ 5 letters, Pansiot words avoiding 3-repetitions form a regular language, which is a rather small superset of the set of all threshold words. Using cylin...

Sequences of Integers Avoiding 3-term Arithmetic Progressions

The optimal length r(n) of a sequence in [1, n] containing no 3term arithmetic progression is determined for several new values of n and some results relating to the subadditivity of r are obtained. We also prove a particular case of a conjecture of Szekeres. A subsequence S = (a1, a2, . . . , ak) of the sequence 〈n〉 = (1, 2, . . . , n) containing no three terms ap, aq, and ar for which aq − ap...