Recently E. Bombieri and N. M. Katz (2010) demonstrated that several well-known results about the distribution of values of linear recurrence sequences lead to interesting statements for Frobenius traces of algebraic curves. Here we continue this line of study and establish the Möbius randomness law quantitatively for the normalised form of Frobenius traces.

We prove a lower bound for the large sieve with square moduli.

In 1984, K. Mahler asked how well elements in the Cantor middle third set can be approximated by rational numbers from that set and by rational numbers outside of that set. We consider more general missing digit sets $C$ and construct numbers in $C$ that are arbitrarily well approximable by rationals in $C$, but badly approximable by rationals outside of $C$. More precisely, we construct them so that all but finitely many of their convergents lie in $C$.

We study transcendence properties of certain infinite products of cyclotomic polynomials. In particular, we determine all cases in which the product is hypertranscendental. We then use various results from Mahler’s transcendence method to obtain algebraic independence results on such functions and their values.

The approximation constant ${\it\lambda}_{k}({\it\zeta})$ is defined as the supremum of ${\it\eta}\in \mathbb{R}$ such that the estimate $\max _{1\leqslant j\leqslant k}\Vert {\it\zeta}^{j}x\Vert \leqslant x^{-{\it\eta}}$ has infinitely many integer solutions $x$. Here $\Vert .\Vert$ denotes the distance to the closest integer. We establish a connection on the joint spectrum $({\it\lambda}_{1}({\it\zeta}),{\it\lambda}_{2}({\it\zeta}),\ldots )$, which will lead to various improvements of known results on the individual spectrum of the approximation constants ${\it\lambda}_{k}({\it\zeta})$ as well. In particular, for given $k\geqslant 1$ and ${\it\lambda}\geqslant 1$, we construct ${\it\zeta}$ in the Cantor set with ${\it\lambda}_{k}({\it\zeta})={\it\lambda}$. Moreover, we establish an estimate for the uniform approximation constants $\widehat{{\it\lambda}}_{k}({\it\zeta})$, which enables us to determine classical approximation constants for Liouville numbers.

We study properties of -free numbers, that is numbers that are not divisible by any member of a set . First we formulate the most-used procedure for finding them (in a given set of integers) as easy-to-apply propositions. Then we use the propositions to consider Diophantine properties of -free numbers and their distribution on almost all short intervals. Results on -free numbers have implications to non-vanishing Fourier coefficients of cusp forms, so this work also gives information about them.

In this paper, we study how close the terms of a finite arithmetic progression can get to a perfect square. The answer depends on the initial term, the common difference and the number of terms in the arithmetic progression.

It is proved that, if $P$ is a polynomial with integer coefficients, having degree 2, and $1>\varepsilon>0$, then $n(n-1)\cdots(n-k+1)=P(m)$ has only finitely many natural solutions $(m,n,k)$, $n\ge k>n\varepsilon$, provided that the $abc$ conjecture is assumed to hold under Szpiro's formulation.