We will talk about recent work showing there are infinitely many primes with no $7$ in their decimal expansion. (And similarly with $7$ replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.[-]

Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal U}(x)^{\lambda}$ contain an interval for all $x\in (0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters associated with the family of symmetric binary expansions studied recently by the first speaker and C. Kalle.
This is joint work with V. Komornik, D. Kong and W. Li.[-]

Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain's strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.[-]

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. [-]

In the first part of the talk we will survey known results concerning Fibonacci numbers whose digital representations in base 10 display some interesting patterns. In the second part of the talk we will give the main steps of the proof of a recent result which states that $b = 4$ is the only integer ≥ 2 such that there are two Fibonacci numbers larger than 1 which are repunits in base b. In this case, $F_{5}=(4^{2}-1)/(4-1)$ and $F_{8}=(4^{3}-1)/(4-1)$. This is joint work with C. A. Gomez and J. C. Gomez from Cali, Colombia.[-]

This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.

In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\][-]

We introduce a family of block-additive automatic sequences, that are obtained by allocating a weight to each couple of digits, and defining the nth term of the sequence as being the total weight of the integer n written in base k. Under an additional combinatorial difference condition on the weight function, these sequences can be interpreted as generalised Rudin–Shapiro sequences. We prove that these sequences have the same two-term correlations as sequences of symbols chosen uniformly and independently at random. The speed of convergence is independent of the prime factor decomposition of k. This extends work by E. Grant, J. Shallit, T. Stoll, and by P.-A. Tahay.[-]