Next Seminar:  June 19, 2024 at TU Wien

Location: Seminarraum AE U1 - 5, Karlsplatz 13

Schedule: 

13:30-14:30: Régis de la Bretèche (Université Paris Cité): "Mean value of Erdős--Hooley Delta-function" 

Abstract:  The  Erdős--Hooley Delta-function is a measure of divisors concentration in a dyadic interval of an integer. Recently, Ford, Koukoulopoulos and Tao proved new upper and lower bound of the mean value of  Erdős--Hooley Delta-function. In a joint work with Tenenbaum, we improve their result. We shall explain the new ideas of  Ford—Koukoulopoulos—Tao and how to improve their results. We shall develop application in counting in diophantine geometry.

14:30-15:00: Tea break

15:00-16:00: Alina Ostafe (UNSW): "On some arithmetic statistics for integer matrices"

Abstract: We consider the set $\mathcal{M}_n(\mathbb{Z}; H))$ of $n\times n$-matrices with integer elements of size at most $H$ and obtain a new upper bound on the number of matrices from $\mathcal{M}_n(\mathbb{Z}; H)$ with a given characteristic polynomial $f \in\mathbb{Z}[X]$, which is uniform with respect to $f$. This complements the asymptotic formula of A. Eskin, S. Mozes and N. Shah (1996) in which $f$ has to be fixed and irreducible. We use our result to address various other questions of arithmetic statistics for matrices from $\mathcal{M}_n(\mathbb{Z}; H)$, eg satisfying certain multiplicative relations. Some of these problems generalise those studied in the scalar case $n=1$ by F. Pappalardi, M. Sha, I. E. Shparlinski and C. L. Stewart (2018) with an obvious distinction due to the non-commutativity of matrices. 

Joint works with Kamil Bulinski, Philipp Habegger and Igor Shparlinski.

16:00-16:15: Break

16:15-17:15: Robert Tichy (TU Graz): "Deterministic Randomness: Correlation and Discrepancy"

Abstract: We study lacunary and polynomial sequences with respect to various measures of pseudorandomness. This includes a comparison of the classical discrepancy with variants of the correlation measure in the sense of Mauduit and Sarkoezy. In particular, we present results for the distribution behaviour of lacunary sequences and for sequences of "almost exponential" growth, such as the so-called Hardy-Littlewood-Polya sequence and sequences satisfying an Erdős gap condition. Furthermore, some results are extended to the general frame of Hardy fields.

18:00: Dinner