We prove that there are absolute constants $c_1\>0$ and $c_2\>0$ such that for every$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$there are$$b_0,b_1,...,b_n\in \{-1,0,1\}$$such that$$P\left(z\right)=\sum _{j=0}^n{b}_j{a}_j{z}^j$$has at least $c_2{n}^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.

In 1955, Roth established that if $\xi$ is an irrational number such that there are a positive real number $\epsilon$ and infinitely many rational numbers $p/q$ with $q\ge 1$ and $|\xi -p/q|\<q^{-2-\epsilon }$, then $\xi$ is transcendental. A few years later, Cugiani obtained the same conclusion with $\epsilon$ replaced by a function $q\mapsto \epsilon \left(q\right)$ that decreases very slowly to zero, provided that the sequence of rational solutions to $|\xi -p/q|\<q^{-2-\epsilon \left(q\right)}$ is sufficiently dense, in a suitable sense. We give an alternative, and much simpler, proof of Cugiani’s Theorem and extend it to simultaneous...

Three theorems of Nagell of 1923 concerning integer values of certain sums of fractions are extended.

We characterize 2-birational CM-extensions of totally real number fields in terms of tame ramification. This result completes in this case a previous work on pro-l-extensions over 2-rational number fields.

We prove that for any real $\theta$ there are infinitely many values of $s=\sigma +it$ with $\sigma \to 1+$ and $t\to +\infty$ such that$$\Re \{exp\left(i\theta \right)logL(s,\chi )\}\ge log\frac{logloglogt}{loglogloglogt}+O\left(1\right).$$The proof relies on an effective version of Kronecker’s approximation theorem.