On some Density Theorems in Number Theory and Group Theory.

摘要

Gowers [31] in his paper on quasirandom groups studies a question of Babai and Sós asking whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|. Answering the question negatively, he proves that for sufficiently large prime p, the group PSL₂($Fp$) has no product-free subset of size ≥ cn 8/9, where n is the order of PSL₂($Fp$).;We will consider the problem for compact groups and in particular for the profinite groups SL_k($Zp$) and Sp_2k($Zp$). In Part I of this thesis, we obtain lower and upper exponential bounds for the supremal measure of the product-free sets. The proof involves establishing a lower bound for the dimension of non-trivial representations of the finite groups SL_k($Z$/(pn$Z$)) and Sp_2k($Z$/(pn$Z$)). Indeed, our theorem extends and simplifies previous work of Landazuri and Seitz [49], where they consider the minimal degree of representations for Chevalley groups over a finite field.;In Part II of this thesis, we move to algebraic number theory. A monogenic polynomial f is a monic irreducible polynomial with integer coefficients which produces a monogenic number field. For a given prime q, using the Chebotarev density theorem, we will show the density of primes p, such that tq – p is monogenic, is greater than or equal to (q – 1)/q. We will also prove that, when q = 3, the density of primes p, which is non-monogenic, is at least 1/9.;Keywords. Profinite group, Complex representation, Hilbert-Schmidt operator, Singular value decomposition, Chebotarev density theorem, Monogenic field, Thue equation..