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operator semigroups . Semigroup Forum 103 , 3 ( 2021 ), 791 – 811 . [5] Babalola , V. A . Semigroups of operators on locally convex spaces . Trans. Am. Math. Soc . 199 ( 1974 ), 163 – 179 . [6] Bátkai , A ., Eisner , T ., and Latushkin , Y
We present examples of multiplicative semigroups of positive reals (Beurling’s generalized integers) with gaps bounded from below.
[1] Arens , R . Operational calculus of linear relations . Pacific J. Math . 11 , 1 ( 1961 ), 9 – 23 . [2] Kato , T . Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups . In Topics in Functional
Asymptotic uniform upper density, shortened as a.u.u.d., or simply upper density, is a classical notion which was first introduced by Kahane for sequences in the real line.
Syndetic sets were defined by Gottschalk and Hendlund. For a locally compact group 𝐺, a set 𝑆 ⊂ 𝐺 is syndetic, if there exists a compact subset 𝐶 ⋐ 𝐺 such that 𝑆𝐶 = 𝐺. Syndetic sets play an important role in various fields of applications of topological groups and semigroups, ergodic theory and number theory. A lemma in the book of Fürstenberg says that once a subset 𝐴 ⊂ ℤ has positive a.u.u.d., then its difference set 𝐴 − 𝐴 is syndetic.
The construction of a reasonable notion of a.u.u.d. in general locally compact Abelian groups (LCA groups for short) was not known for long, but in the late 2000’s several constructions were worked out to generalize it from the base cases of ℤ𝑑 and ℝ𝑑. With the notion available, several classical results of the Euclidean setting became accessible even in general LCA groups.
Here we work out various versions in a general locally compact Abelian group 𝐺 of the classical statement that if a set 𝑆 ⊂ 𝐺 has positive asymptotic uniform upper density, then the difference set 𝑆 − 𝑆 is syndetic.
In this paper we work out a Riemann–von Mangoldt type formula for the summatory function