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Characterizations of groups generated by Kronecker sets

András Biró — 2007

Journal de Théorie des Nombres de Bordeaux

In recent years, starting with the paper [B-D-S], we have investigated the possibility of characterizing countable subgroups of the torus $T = R / Z$ by subsets of $Z$ . Here we consider new types of subgroups: let $K \subseteq T$ be a Kronecker set (a compact set on which every continuous function $f : K \to T$ can be uniformly approximated by characters of $T$ ), and $G$ the group generated by $K$ . We prove (Theorem 1) that $G$ can be characterized by a subset of $Z^{2}$ (instead of a subset of $Z$ ). If $K$ is finite, Theorem 1 implies our earlier result...

We solve unconditionally the class number one problem for the 2-parameter family of real quadratic fields ℚ(√d) with square-free discriminant d = (an)²+4a for positive odd integers a and n.

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Characterizations of groups generated by Kronecker sets

Yokoi's conjecture

Divisibility of integer polynomials and tilings of the integers

Chowla's conjecture

The class number one problem for the real quadratic fields ℚ(√((an)²+4a))