Nested ideals and topologically u I-torsion elements of the circle group
Abstract
Let u=(un)n∈ N be a sequence in N+ with u0=1 and un un+1 for every n∈ N, and let bn:=un+1/un for every n∈ N+. For every r∈ [0,1), there exists a unique sequence (cn)n∈ N+ in N such that r= Σn=1∞cnun, with cn<bn for every n∈ N+, and cn<bn-1 for infinitely many n∈ N+; let supp(r):=\n∈ N+: cn≠0\ and suppb(r):=\n∈ N+: cn = bn-1\. For x=r+ Z∈ T, let supp(x) = supp(r) and suppb(x) = suppb(r). For an ideal I of N, an element x of the circle group T is called a topologically u I-torsion element of T if unx I-converges to 0, that is, \n∈ N: unx ∈ U\∈ I for every neighborhood U of 0 in T. In this paper, under suitable conditions on the ideal I, we completely describe the u I-torsion elements x of T with n∈supp(x)bn=∞ and those with \bn:n∈supp(x)\ bounded. According to Corollary 2.12 in [A. Ghosh, Ric. Mat. 73 (2024), 2263--2281], an element x ∈ T with \bn:n∈supp(x)\ bounded is topologically u I-torsion if and only if supp(x)+1 supp(x)∈ I and supp(x) suppb(x) ∈ I. We characterize the ideals I of N, naming them nested, such that this equivalence holds and we provide examples of non-nested ideals I that satisfy the above mentioned suitable conditions, so that the equivalence claimed by Ghosh fails for those I.
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