Mathematics Colloquium
Speaker: Jean-Marc Deshouillers, University of Bordeaux
Host: Keshav Aggarwal
Title: Coprimality of consecutive elements in regular sequences
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 28
Venue: Ramanujan Hall
Abstract: ``Regular sequences'' are integral sequences $ \mathcal{A}_f$ of the shape $\{ \lfloor f(n) \rfloor, n \in \mathbb{N}\}$, where $f$ is a smooth function from $[1, + \infty)$ to itself with polynomial growth (for example the Segal-Piatetski-Shapiro sequences where $f(n)=n^c$). We shall study the following questions: are there infinitely many arbitrarily large blocks of consecutive elements of $ \mathcal{A}_f$ which are pairwise coprime? If so, do they occur often? To illustrate the results in the case of SPS sequences, we show that the sequence $SPS(c)$ contains arbitrary large blocks of pairwise coprime elements when $1<c<2$. Moreover, those blocks of given length occur with positive density. For $c>2$ although we expect the same result to hold, we can prove it only for $c$ rational less than $111/55$. Proofs involve combinatorial constructions and harmonic analysis.\\ \textit{From joint works with M. Drmota and C. M\"{u}llner, with S. Naik and a work in progress with H. Iwaniec.}