منابع مشابه
Characterized Subgroups of Topological Abelian Groups
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class o...
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Let X be a compact metrizable abelian group and u = {un} be a sequence in its dual X. Set su(X) = {x : (un, x) → 1} and T H 0 = {(zn) ∈ T ∞ : zn → 1}. Let G be a subgroup of X. We prove that G = su(X) for some u iff it can be represented as some dually closed subgroup Gu of ClXG × T H 0 . In particular, su(X) is polishable. Let u = {un} be a T -sequence. Denote by (X̂,u) the group X ∧ equipped w...
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We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich’s theorem saying that every proper analytic subgroup of the r...
متن کاملOn T-Characterized Subgroups of Compact Abelian Groups
A sequence {un}n∈ω in abstract additively-written Abelian group G is called a T -sequence if there is a Hausdorff group topology on G relative to which limn un = 0. We say that a subgroupH of an infinite compact Abelian groupX is T -characterized if there is a T -sequence u = {un} in the dual group of X , such that H = {x ∈ X : (un, x) → 1}. We show that a closed subgroupH ofX is T -characteriz...
متن کاملSets characterized by missing sums and differences
A more sums than differences (MSTD) set is a finite subset S of the integers such |S + S| > |S− S|. We show that the probability that a uniform random subset of {0, 1, . . . , n} is an MSTD set approaches some limit ρ > 4.28× 10−4. This improves the previous result of Martin and O’Bryant that there is a lower limit of at least 2 × 10−7. Monte Carlo experiments suggest that ρ ≈ 4.5× 10−4. We pre...
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ژورنال
عنوان ژورنال: Topology and its Applications
سال: 2017
ISSN: 0166-8641
DOI: 10.1016/j.topol.2017.08.017