On some Density Theorems in Number Theory and Group Theory
Gowers in his paper on quasirandom groups studies a question of Babai and Sos 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 $\mathrm{PSL}_2(\mathbb{F}_p)$ has no product-free subset of size $\geq cn^{8/9}$, where $n$ is the order of $\mathrm{PSL}_2(\mathbb{F}_p)$.   We will consider the problem for compact groups and in particular for the profinite groups $\SL_k(\mathh{Z}_p)$ and $\Sp_{2k}(\mathbb{Z}_p)$.  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 $\SL_k(\mathbb{Z}/(p^n\mathbb{Z}))$ and $\Sp_{2k}(\mathbb{Z}/(p^n\mathbb{Z}))$.  Indeed, our theorem extends and simplifies previous work of Landazuri and Seitz, 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 $t^q-p$ 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 $\mathbb{Q}(\sqrt[3]{p})$ is non-monogenic, is at least $1/9$.
Profinite group Complex representation Hilbert-Schmidt operator Singular value decomposition Chebotarev density theorem Monogenic field Thue equation