An example of a finite number of recursively enumerable m-degrees containing an infinite sequence of recursively enumerable sets - Bojkova, Elka (Ann. Sofia Univ. Fac. Math. and Inf., 1995)
The aim of this paper is to construct for an arbitrary natural number n a recursively enumerable (r.e.) set A such that the sets A,A^2...,A^n belong to different m-degrees and the sets A^n, A^n+1, A^n+2,.. belong to the same m-degree. That will provide an example of n distinct r.e. m-degrees d_m(A), d_m(A^2),..,d_m(A^n) belonging to one r.e. btt-degree (more precisely, c-degree, the degree d_c(A) of the set A coincidental with d_c(A^2),..,d_c(A^n),..). It suffices to prove that the set A^n+1 is m-reducible to the set A^n-1. For this purpose we construct a scheme for building a r.e. set A, corresponding to a sequence of r.e. sets A_0,A_1,A_2,.., such that the first condition holds regardless the choise of A_0,A_1,A_2,..,. Further we build the sets A_0,A_1,A_2,.., by steps, using the priority arguments to ensure the second condition.