Unrecognizable Sets of Numbers

Abstract

When is a set A of positive integers, represented as binary numbers, "regular" in the sense that it is a set of sequences that can be recognized by a finite-state machine? Let pie A(n) be the number of members of A less than the integer n. It is shown that the asymptotic behavior of pie A(n) is subject to severe restraints if A is regular. These constraints are violated by many important natural numerical sets whose distribution functions can be calculated, at least asymptotically. These include the set P of prime numbers for which pie P(n)~n/log n for large n, the set of integers A (k) of the form n to the power k for which pie A(k)(n)1/k, and many others. The technique cannot, however, yield a decision procedure for regularity since for every infinite regular set A there is a nonregular set A for which /pie Z(n)-pie A(n)/is less than or equal to 1, so that the asymptotic behaviors of the two distribution functions are essentially identical.