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SUMMARY:Piotr Miska (Jagiellonian University)
DTSTART:20220404T121500Z
DTEND:20220404T131500Z
DTSTAMP:20260423T041526Z
UID:WarsawNT/41
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/WarsawNT/41/
 ">(R)-dense and (N)-dense subsets of positive integers and generalized quo
 tient sets</a>\nby Piotr Miska (Jagiellonian University) as part of Warsaw
  Number Theory Seminar\n\nLecture held in currently online (via Zoom).\n\n
 Abstract\nA subset $A$  of the set of positive integers is (R)-dense if it
 s quotient set $ R(A)=\\{a/b \\colon a\, b \\in A\\}$ is dense in the posi
 tive real half-line (with respect to natural topology on real numbers). It
  is a classical result that the set of prime numbers is (R)-dense. The pro
 of of this fact is based on the property of counting function of prime num
 bers. Actually\, this proof shows something more. Namely\, for each infini
 te subset $B$ of the set of positive integers\, the set $R(P\,B)=\\{p/b \\
 colon p \\in P\, b \\in B\\}$ is dense in the set of positive real numbers
 . This motivates to introduce the notion of (N)-denseness. We say that a s
 et $A$ of positive integers is (N)-dense if the set $R(A\,B)$ is dense in 
 the set of positive real numbers for every set $B$ of positive integers. D
 uring the talk we will consider characterizations of (N)-dense sets and co
 nnections between (N)-denseness of a given set.\nIn 2019 Leonetti and Sann
 a introduced the notion of direction sets $D^k(A)=\\{(a_1/\\|a\\|^2\, \\ld
 ots\, a_k/\\|a\\|^2)\\colon a=(a_1\,\\ldots\, a_k) \\in A^k\\}$ that allow
 s us to generalize the property of (R)-denseness. Indeed\, $A$ is (R)-dens
 e if and only if $D^2(A)$ is dense in the set of points of unit circle wit
 h all the coordinates positive. We will see that denseness of $D^k(A)$ in 
 the set of points of unit sphere with all the coordinates positive is equi
 valent to denseness of the generalized quotient set $R^k(A)=\\{(a_1/a_k\,\
 \ldots\, a_{k-1}/a_k)\\colon a_1\,\\ldots\, a_k \\in A\\}$ in the set of p
 oints of $R^{k-1}$ with all the coordinates positive.\nWe will also show s
 ome connections between (N)-denseness of a given set $A$ and denseness of 
 sets $R^k(A)$ with the counting function of $A$ and its dispersion.\nThe t
 alk is based on a joint work with János T. Tóth.\n
LOCATION:https://researchseminars.org/talk/WarsawNT/41/
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