Let $S$ be a fixed symmetric finite subset of $S{L}_d\left({𝒪}_K\right)$ that generates a Zariski dense subgroup of $S{L}_d\left({𝒪}_K\right)$ when we consider it as an algebraic group over $mathbbQ$ by restriction of scalars. We prove that the Cayley graphs of $S{L}_d({𝒪}_K/I)$ with respect to the projections of $S$ is an expander family if $I$ ranges over square-free ideals of ${𝒪}_K$ if $d=2$ and $K$ is an arbitrary numberfield, or if $d=3$ and $K=\mathbb{Q}$.

We note that every positive integer N has a representation as a sum of distinct members of the sequence ${d(n!)}_{n\ge 1}$, where d(m) is the number of divisors of m. When N is a member of a binary recurrence $u={uₙ}_{n\ge 1}$ satisfying some mild technical conditions, we show that the number of such summands tends to infinity with n at a rate of at least c₁logn/loglogn for some positive constant c₁. We also compute all the Fibonacci numbers of the form d(m!) and d(m₁!) + d(m₂)! for some positive integers m,m₁,m₂.

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{U_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)/\left(U_{d^k}\right))$ (i=1,...,m) or ${\prod }_{\genfrac{}{}{0pt}{}{k=1}{V_{d^k}\ne -a_i}}^{\infty }(1+\left(a_i\right)\left(V_{d^k}\right)$ (i=1,...,m) to be algebraically dependent, where $a_i$ are non-zero integers and $U_n$ and $V_n$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_1,...,a_m$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....

Let $X/K$ be an absolutely simple abelian variety over a number field; we study whether the reductions $X_{𝔭}$ tend to be simple, too. We show that if $End\left(X\right)$ is a definite quaternion algebra, then the reduction $X_{𝔭}$ is geometrically isogenous to the self-product of an absolutely simple abelian variety for $𝔭$ in a set of positive density, while if $X$ is of Mumford type, then $X_{𝔭}$ is simple for almost all $𝔭$. For a large class of abelian varieties with commutative absolute endomorphism ring, we give an explicit upper bound...

We give an explicit construction of an integral basis for a radical function field $K=k(t,\rho )$, where ${\rho }^n=D\in k\left[t\right]$, under the assumptions $[K:k(t\left)\right]=n$ and $char\left(k\right)\nmid n$. The field discriminant of $K$ is also computed. We explain why these questions are substantially easier than the corresponding ones in number fields. Some formulae for the $P$-signatures of a radical function field are also discussed in this paper.

We extend Champernowne’s construction of normal numbers to base b to the ${\mathbb{Z}}^d$ case and obtain an explicit construction of a generic point of the ${\mathbb{Z}}^d$ shift transformation of the set ${0,1,...,b-1}^{{\mathbb{Z}}^d}$.