A powerful abelian square-free substitution over 4 letters

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 , where and are permutations (anagrams) of each other. The case of the four letter alphabet Σ4={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 g85 of . The size of this g85, i.e., , is equal to 4×85 (uniform modulus). Until recently, all known methods for constructing arbitrarily long a-2-free words on Σ4 have been based on the structure of g85 and on the endomorphism g98 of found in 2002.In this paper, a great many new a-2-free endomorphisms of 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 , can be used to construct an a-2-free (commutatively functional) substitution σ109 of with 12 image words for each letter.The properties of σ109 lead to a considerable improvement for the lower bound of the exponential growth of cn, i.e., of the number of a-2-free words over 4 letters of length . It is obtained that cn>β−50βn with β=121/m≃1.02306. Originally, in 1998, Carpi established the exponential growth of cn by showing that cn>β−tβn with β=219/t=219/(853−85)≃1.000021, where is the modulus of the substitution that he constructs starting from g85.