Abundance of progressions in a commutative semigroup by elementary means
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Abundance of progressions in a commutative semigroup by elementary means Sayan Goswami1 · Subhajit Jana1 Received: 22 July 2019 / Accepted: 11 September 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Furstenberg and Glasner proved that for an arbitrary k ∈ N, any piecewise syndetic set contains a k-term arithmetic progression and such collection is also piecewise syndetic in Z. They used the algebraic structure of βN. The above result was extended for arbitrary semigroups by Bergelson and Hindman, again using the structure of the ˇ Stone–Cech compactification of a general semigroup. Beiglböck provided an elementary proof of the above result and asked whether the combinatorial argument in his proof can be enhanced in a way which makes it applicable to a more abstract setting. In this work we extend that technique of Beiglböck in commutative semigroups. ˇ Keywords Semigroup · Stone–Cech compactification · Syndetic
1 Introduction A subset S of Z is called syndetic if there exists r ∈ N such that ri=1 (S − i) = Z. Again a subset S of Z is called thick if it contains arbitrarily long intervals. Sets which can be expressed as the intersection of thick and syndetic sets are called piecewise syndetic. For a general commutative semigroup (S, +), a set A ⊆ S issaid to be syndetic in (S, +), if there exist a finite nonempty set F ⊆ S such that t∈F (−t + A) = S where −t + A = {s ∈ S : t + s ∈ A}. A set A ⊆ S is said to be thick if for every finite nonempty set E ⊆ S, there exists an element x ∈ S such that E + x ⊆ A. A set A ⊆ S is said to be piecewise syndetic set if there exist a finite nonempty set F ⊆ S
Communicated by Jimmie D. Lawson.
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Sayan Goswami [email protected] Subhajit Jana [email protected]
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Department of Mathematics, University of Kalyani, Kalyani, Nadia, West Bengal 741235, India
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S. Goswami, S. Jana
such that t∈F (−t + A) is thick in S. It can be proved that a piecewise syndetic set is the intersection of a thick set and a syndetic set [6, Theorem 4.49]. One of the famous Ramsey theoretic results is van der Waerden’s Theorem [7] which states that, given r , k ∈ N there is some l ∈ N such that one cell of any partition {C1 , C2 , . . . , Cr } of {1, 2, . . . , l} contains an arithmetic progressions of length k. It follows from van der Waerden’s Theorem that any piecewise syndetic subset A of N contains arbitrarily long arithmetic progressions. To see this, pick finite F ⊆ N such that t∈F (−t + A) is thick in N. Let r = |F| and let a length kbe given. Pick l as guaranteed for r and k and pick x such that {1, 2, . . . , l} + x ⊆ t∈F (−t + A). For t ∈ F, let Ct = {y ∈ {1, 2, . . . , l} : y + x ∈ (−t + A)}. Pick a and d in N and t ∈ F such that {a, a + d, a + 2d, . . . , a + (k − 1)d} ⊆ Ct and let a = a + x + t. Then {a , a + d, a + 2d, . . . , a + (k − 1)d} ⊆ A. A homothetic copy of a finite set F in a commutative semigroup (S, +), is of the form a + n · F = {a + n · x : x ∈ F}, where a ∈ S, n ∈ N and n · x is the sum of x with itself n times
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