| 研究実績の概要 |
Let n be a positive integer. We study the automorphism group Aut(G) of a dense subgroup G of R^n. We show that Aut(G) can be naturally identified with the subgroupΦ(G) of the group GL(n,R) of all all non-degenerated square n-matrices A with real coefficients such that G A = G. We describe Φ(G) for many dense subgroups G of either the real line R or the plane R^2. We consider also an inverse problem of which symmetric subgroups of GL(n,R) can be realized as Φ(G) for some dense subgroup G of R^n.
For n>=2, we show that any proper subgroup H of GL(n,R) satisfying SO(n,R) ⊆ H cannot be realized in this way. (Here SO(n,R) denotes the special orthogonal group of dimension n.) We show that the realization problem is quite non-trivial even in the one-dimensional case and has deep connections to number theory.
For a positive integer n, we investigate subsets of R^n which generate it by the use of positive integers taken as multipliers, as well as a related question of which sub-semigroups of R^n are generated by different subsets of R^n. In particular, we characterize sigma-compact subsets of R^n generating it in this way, and show that this characterization does not hold for general subsets.
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