Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

Collection Jean-Morlet Chair 2020 - Conference: Diophantine Problems, Determinism and Randomness / Chaire Jean-Morlet 2020 - Conférence : Problèmes diophantiens, déterminisme et aléatoire

Organizer(s) Rivat, Joël ; Tichy, Robert

Date(s) 23/11/2020 - 27/11/2020

linked URL https://www.chairejeanmorlet.com/2256.html

00:00:00 / 00:00:00

3 28

(Logarithmic) densities for automatic sequences along primes and squares

By Michael Drmota

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.

Information about the video

Date of recording 23/11/2020
Date of publication 01/12/2020
Institution CIRM
Licence CC BY NC ND
Language English
Audience Researchers
Director(s) Guillaume Hennenfent
Format MP4

Citation data

DOI 10.24350/CIRM.V.19686703
Cite this video Drmota, Michael (23/11/2020). (Logarithmic) densities for automatic sequences along primes and squares. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19686703
URL https://dx.doi.org/10.24350/CIRM.V.19686703

Domain(s)

Number Theory

Bibliography

ADAMCZEWSKI, Boris, DRMOTA, Michael, et MÜLLNER, Clemens. (Logarithmic) densities for automatic sequences along primes and squares. arXiv preprint arXiv:2009.14773, 2020. - https://arxiv.org/abs/2009.14773
MAUDUIT, Christian, RIVAT, Joël, et al. La somme des chiffres des carrés. Acta Mathematica, 2009, vol. 203, no 1, p. 107-148. - http://dx.doi.org/10.1007/s11511-009-0040-0
MAUDUIT, Christian et RIVAT, Joël. Sur un probleme de Gelfond: la somme des chiffres des nombres premiers. Annals of Mathematics, 2010, p. 1591-1646. - https://www.jstor.org/stable/20752248

MSC codes

Last related questions on MathOverflow

You have to connect your Carmin.tv account with mathoverflow to add question

Ask a question on MathOverflow

All the collection videos

49:22

published on December 1, 2020

Classification and statistics of cut-and-project sets

By Barak Weiss

51:02

published on December 1, 2020

On binary quartic Thue equations and related topics

By Gary Walsh

46:04

published on December 1, 2020

(Logarithmic) densities for automatic sequences along primes and squares

By Michael Drmota

58:16

published on December 1, 2020

Independence of actions of (N,+) and (N,×) and Sarnak's Möbius disjointness conjecture

By Vitaly Bergelson

51:06

published on December 1, 2020

Higher moments of primes in intervals and in arithmetic progressions, II

By Régis De la Bretèche

48:41

published on December 1, 2020

$D(n)$-sets with square elements

By Andrej Dujella

57:00

published on December 1, 2020

Diophantine exponents, best approximation and badly approximable numbers

By Nikolay Moshchevitin

53:35

published on December 1, 2020

Fibonacci numbers and repdigits

By Florian Luca

51:38

published on December 1, 2020

Dynamical irreducibility of polynomials modulo primes

By Alina Ostafe

45:01

published on December 1, 2020

Number of solutions to a special type of unit equations in two unknowns

By István Pink

48:07

published on December 1, 2020

On S-Diophantine Tuples

By Volker Ziegler

46:24

published on December 1, 2020

Large values of the remainder term of the prime number theorem

By János Pintz

52:52

published on December 1, 2020

Improved cap constructions, and sets without arithmetic progressions

By Christian Elsholtz

48:47

published on December 1, 2020

On generalised Rudin-Shapiro sequences

By Thomas Stoll

44:56

published on December 1, 2020

Constructing abelian extensions with prescribed norms

By Christopher Frei

41:07

published on December 1, 2020

The Rudin-Shapiro function in finite fields

By Cécile Dartyge

47:08

published on December 1, 2020

Effective finiteness results for diophantine equations over finitely generated domains

By Kalman Györy

51:40

published on December 1, 2020

Modularity of the q-Pochhammer symbol and application

By Sary Drappeau

52:28

published on December 1, 2020

Skolem's conjecture and exponential Diophantine equations

By Lajos Hajdu

47:59

published on December 1, 2020

The sum-of-digits function in linearly recurrent number systems and almost primes

By Manfred Madritsch

$Some interactions between number theory and multifractal analysis$
53:28

published on December 1, 2020

Some interactions between number theory and multifractal analysis

By Bruno Martin

$Multidimensional continued fractions and symbolic codings of toral translations$
57:17

published on December 1, 2020

Multidimensional continued fractions and symbolic codings of toral translations

By Jörg Thuswaldner

56:48

published on December 1, 2020

Poisson-generic points

By Benjamin Weiss

46:39

published on December 1, 2020

On some diophantine equations in separated variables

By Attila Bérczes

55:27

published on December 1, 2020

Equidistribution of roots of unity and the Mahler measure

By Philipp Habegger

49:16

published on December 1, 2020

Bertini and Northcott

By Fabien Pazuki

53:20

published on December 1, 2020

Zaremba's conjecture and growth in groups

By Ilya Shkredov

51:44

published on December 1, 2020

Pseudorandomness at prime times and digits of Mersenne numbers

By Igor Shparlinski

Copyright Carmin.tv 2024

Give feedback