(R)-dense and (N)-dense subsets of positive integers and generalized quotient sets

Abstract: A subset $A$ of the set of positive integers is (R)-dense if its quotient set $ R(A)=\{a/b \colon a, b \in A\}$ is dense in the positive 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 proof of this fact is based on the property of counting function of prime numbers. Actually, this proof shows something more. Namely, for each infinite 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 set $A$ 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. During the talk we will consider characterizations of (N)-dense sets and connections between (N)-denseness of a given set. In 2019 Leonetti and Sanna introduced the notion of direction sets $D^k(A)=\{(a_1/\|a\|^2, \ldots, a_k/\|a\|^2)\colon a=(a_1,\ldots, a_k) \in A^k\}$ that allows us to generalize the property of (R)-denseness. Indeed, $A$ is (R)-dense if and only if $D^2(A)$ is dense in the set of points of unit circle with 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 equivalent 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 points of $R^{k-1}$ with all the coordinates positive. We will also show some 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. The talk is based on a joint work with János T. Tóth.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar