Abundance of arithmetic progressions in ๐โ-sets
arXiv (Cornell University)๏ผ2022๏ผ
Abstract
H.Furstenberg and E.Glasner proved that for an arbitrary kโโ, any piecewise syndetic set of integers contains a k-term arithmetic progression and the collection of such progressions is itself piecewise syndetic in โค. The above result was extended for arbitrary semigroups by V. Bergelson and N. Hindman, using the algebra of the Stone-ฤech compactification of discrete semigroups. However, they provided an abundance for various types of large sets. In , the first author, Neil Hindman and Dona Strauss introduced two notions of large sets, namely, J-set and C-set. In , V. Bergelson and D. Glasscock introduced another notion of largeness, which is analogous to the notion of J-set, namely ๐โ- set. All these sets contain arithmetic progressions of arbitrary length. In , the second author and S. Goswami proved that for any J-set, Aโโ, the collection {(a,b): {a,a+b,a+2b,โฆ,a+lb}โ A} is a J-set in (โรโ,+). In this article, we prove the same for ๐โ-sets.
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Key words
arithmetic progression
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