Related papers: Effective bounds for Roth's theorem with shifted s…

We consider a random walk $(Y_N)_{N\geq 0}$ on $\mathbb{R}^2$ generated by successively applying independent random isometries, drawn from a fixed measure $\mu$, to the point $0$. When the support of $\mu$ is finite and includes an…

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th…

For most purposes, one can replace the use of Rolle's theorem and the mean value theorem, which are not constructively valid, by the law of bounded change. The proof of two basic results in numerical analysis, the error term for Lagrange…

Under reasonable algebraic assumptions and under an infinite second order moment assumption, we show that the logarithm of the norm (log-norm) of a product of random i.i.d. matrices with entries in $\mathbb{R}$ or in any other local field…

In this note, we consider Szemer\'{e}di's theorem on $k$-term arithmetic progressions over finite fields $\mathbb{F}_p^n$, where the allowed set $S$ of common differences in these progressions is chosen randomly of fixed size. Combining a…

It is shown that an arbitrary function from $D\subset \R^n$ to $\R^m$ will become $C^{0,\alpha}$-continuous in almost every $x\in D$ after restriction to a certain subset with limit point $x$. For $n\geq m$ differentiability can be…

We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…

A great challenge in the analysis of the discrepancy function D_N is to obtain universal lower bounds on the L-infty norm of D_N in dimensions d \geq 3. It follows from the average case bound of Klaus Roth that the L-infty norm of D_N is at…

This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0…

The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is…

In this article, we give explicit bounds of order $\log t$ for $\sigma$ close to $1$, for two quantities: $|\zeta'(\sigma +it)/\zeta(\sigma +it)|$ and $|1/\zeta(\sigma +it)|$. We correct an error in the literature, and especially in the…

We give a level one result for the "symmetry integral", say $I_f(N,h)$, of essentially bounded $f:\N \to \R$; i.e., we get a kind of "square-root cancellation" \thinspace bound for the mean-square (in $N<x\le 2N$) of the "symmetry"…

Let b > 1 be an integer and denote by s_b(m) the sum of the digits of the positive integer m when is written in base b. We prove that s_b(n!) > C_b log n log log log n for each integer n > e, where C_b is a positive constant depending only…

Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled centered entries are uniformly square integrable. We prove that if the entries of $X_N$ can be partitioned into independent subsets each of size $o(\log N)$, then…

Let $a \geq 2$ be an integer. We prove that for every periodic sequence $(s_n)_{n \geq 1}$ in $\{-1, +1\}$ there exists an effectively computable rational number $C_\mathbf{s} > 0$ such that \begin{equation*} \log\operatorname{lcm}(a + s_1,…

We prove a central limit theorem for Birkhoff sums of the Rosen continued fraction algorithm. A Lasota-Yorke bound is obtained for general one-dimensional continued fractions with the bounded variation space, which implies quasi-compactness…

Let the continued fraction expansion of any irrational number $t \in (0,1)$ be denoted by $[0,a_{1}(t),a_{2}(t),...]$ and let the i-th convergent of this continued fraction expansion be denoted by $c_{i}(t)/d_{i}(t)$. Let \[ S=\{t \in…

We consider a conservative ergodic measure-preserving transformation $T$ of the measure space $(X,\mathcal{B},\mu)$ with $\mu$ a $\sigma$-finite measure and $\mu(X)=\infty$. Given an observable $g:X\to \mathbb{R}$, it is well known from…

Oort-Zink proved that a $p$-divisible group over a normal base in characteristic $p$ with constant Newton polygon is isogenous to a $p$-divisible group admitting a slope filtration. In this paper, we generalize this result to log…

Consider the sequence of continued fraction convergents $p_n/q_n$ to a random irrational number. We study the distribution of the sequences $p_n \pmod{m}$ and $q_n \pmod{m}$ with a fixed modulus $m$, and more generally, the distribution of…