A powerful abelian square-free substitution over 4 letters

Abstract

In 1961, Paul Erdös posed the question whether abelian squares can be avoided in arbitrarily long words over a finite alphabet. An abelian square is a non-empty word uv , where u and v are permutations (anagrams) of each other. The case of the four letter alphabet Σ 4 = { a , b , c , d } turned out to be the most challenging and remained open until 1992 when the author presented an abelian square-free (a-2-free) endomorphism g 85 of Σ 4 ∗ . The size of this g 85 , i.e., | g 85 ( abcd ) | , is equal to 4 × 85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ 4 have been based on the structure of g 85 and on the endomorphism g 98 of Σ 4 ∗ found in 2002. In this paper, a great many new a-2-free endomorphisms of Σ 4 ∗ are reported. The sizes of these endomorphisms range from 4 × 102 to 4 × 115 . Importantly, twelve of the new a-2-free endomorphisms, of modulus m = 109 , can be used to construct an a-2-free (commutatively functional) substitution σ 109 of Σ 4 ∗ with 12 image words for each letter. The properties of σ 109 lead to a considerable improvement for the lower bound of the exponential growth of c n , i.e., of the number of a-2-free words over 4 letters of length n . It is obtained that c n > β − 50 β n with β = 1 2 1 / m ≃ 1.02306 . Originally, in 1998, Carpi established the exponential growth of c n by showing that c n > β − t β n with β = 2 19 / t = 2 19 / ( 8 5 3 − 85 ) ≃ 1.000021 , where t = 8 5 3 − 85 is the modulus of the substitution that he constructs starting from g 85 .