A Diophantine equation has separated variables if it is of the form $ f(x) = g(y)$ for polynomials $f$, $g$. In a more general sense the degree of $f $ and $g$ may also be a variable.In the present talk various results for special types of the polynomials $f$ and $g$ will be presented. The types of the considered polynomials contain power sums, sums of products of consecutive integers, Komornik polynomials, perfect powers. Results on $F$-Diophantine sets, which are proved using results on Diophantine equations in separated variables will also be considered. The main tool for the proof of the presented general qualitative results is the famous Bilu-Tichy Theorem. Further, effective results (which depend on Baker's method) and results containing the complete solutions to special cases of these equations will also be included.[-]

We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdös-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak's Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,x). The talk is based on recent joint work with Florian Richter.[-]

Let $q=p^r$, where $p$ is a prime number and $ ß=(\beta_1 ,\ldots ,\beta_r)$ be a basis of $\mathbb{F}_q$ over $\mathbb{F}_p$.
Any $\xi \in \mathbb{F}_q$ has a unique representation $\xi =\sum_{i=1}^r x_i \beta _i$ with $x_1,\ldots ,x_r \in \mathbb{F}_p$.
The coefficients $x_1,\ldots ,x_r$ are called the digits of $\xi$ with respect to the basis $ß$.
The analog of the Rudin-Shapiro function is $R(\xi)=x_1x_2+\cdots + x_{r-1}x_r$. For $f \in \mathbb{F}_q [X]$, non constant and $c\in\mathbb{F}_p$, we obtain some formulas for the number of solutions in $\mathbb{F}_q$ of $R(f(\xi ))=c$. The proof uses the Hooley-Katz bound for the number of zeros of polynomials in $\mathbb{F}_p$ with several variables.

This is a joint work with László Mérai and Arne Winterhof.[-]

It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two breakthroughresults of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes thesequence of primes). More technically, one has to develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. Then asan application one can deduce that the logarithmic densities of any automatic sequence along squares $(n^2){n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, if densities exist then they are (usually) rational. [-]

For an integer n, a set of distinct nonzero integers $\left \{ a_{1},a_{2},...a_{m} \right \}$ such that $a_{i}a_{j}+n$ is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property $D(n)$ or simply a $D(n)$-set. $D(1)$-sets are known as Diophantine m-tuples. When considering $D(n)$-sets, usually an integer n is fixed in advance. However, we may ask if a set can have the property $D(n)$ for several different n's. For example, {8, 21, 55} is a $D(1)$-triple and $D(4321)$-triple. In a joint work with Adzaga, Kreso and Tadic, we presented several families of Diophantine triples which are $D(n)$-sets for two distinct n's with $n\neq 1$. In a joint work with Petricevic we proved that there are infinitely many (essentially different) quadruples which are simultaneously $D(n_{1})$-quadruples and $D(n_{2})$-quadruples with $n_{1}\neq n_{2}$. Morever, the elements in some of these quadruples are squares, so they are also $D(0)$-quadruples. E.g. $\left \{ 54^{2}, 100^{2}, 168^{2}, 364^{2}\right \} $ is a $D(8190^{2})$, $D(40320^{2})$ and $D(0)$-quadruple. In this talk, we will describe methods used in constructions of mentioned triples and quadruples. We will also mention a work in progress with Kazalicki and Petricevic on $D(n)$-quintuples with square elements (so they are also $D(0)$-quintuples). There are infinitely many such quintuples. One example is a $D(4804802)$-quintuple $\left \{ 225^{2}, 286^{2}, 819^{2}, 1408^{2}, 2548^{2}\right \}$.[-]

In the 1980's we developed an effective specialization method and used it to prove effective finiteness theorems for Thue equations, decomposable form equations and discriminant equations over a restricted class of finitely generated domains (FGD's) over $\mathbb{Z}$ which may contain not only algebraic but also transcendental elements. In 2013 we refined with Evertse the method and combined it with an effective result of Aschenbrenner (2004) concerning ideal membership in polynomial rings over $\mathbb{Z}$ to establish effective results over arbitrary FGD's over $\mathbb{Z}$. By means of our method general effective finiteness theorems have been obtained in quantitative form for several classical Diophantine equations over arbitrary FGD's, including unit equations, discriminant equations (Evertse and Gyory, 2013, 2017), Thue equations, hyper- and superelliptic equations, the Schinzel–Tijdeman equation (Bérczes, Evertse and Gyory, 2014), generalized unit equations (Bérczes, 2015), and the Catalan equation (Koymans, 2015). In the first part of the talk we shall briefly survey these results. Recently we proved with Evertse effective finiteness theorems in quantitative form for norm form equations, discriminant form equations and more generally for decomposable form equations over arbitrary FGD's. In the second part, these new results will be presented. Some applications will also be discussed.[-]

Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials, extending a result of Lind, Schmidt, and Verbitskiy. Our method draws from earlier work of Duke.[-]