UC Irvine

In this thesis, we study expansions of the additive group of integers (Z, +). In particular, weare interested in expansions of the group structure on Z which preserve some of the model- theoretic properties of the group. For example, it is well known that (Z, +) is superstable of Lascar rank 1. Much work has been done to determine which expansions of (Z, +) by a predicate yield a structure which is still (super)stable. Relatedly, one can ask about which predicates yield expansions that, while unstable, are still (super)simple. Using notions from algorithmic randomness, we show that for almost all subsets A of the natural numbers, the expansion of (Z, +) by the set ±A = {a ∈ Z : |a| ∈ A} is unstable, supersimple, and has Lascar rank 1, providing one answer to a question posed by Bhardwaj and Tran. Additionally, we show how the expansions of (Z, +) by the semiprimes (products of two primes) and separately the quasiprimes (numbers with no small prime factors) also satisfy some similar classification results by proving that these sets behave generically with respect to the additive structure on Z. In the case of the semiprimes, this comes from assuming an unproven number-theoretic result similar to Dickson’s conjecture for the prime numbers. In the case of the quasiprimes, we use results from sieve theory to derive the needed genericity.