In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.

A set is called piecewise syndetic if there exists a finite subset G of such that for every finite subset F of there exists an such that x + F ⊂ ⋃ n ∈ G ( S − n ) {\displaystyle x+F\subset \bigcup _{n\in G}(S-n)}

2015年12月3日 · A syndetic set $S$ is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded.

2024年8月23日 · Our main result is a characterization of dynamically central syndetic sets as precisely those sets that belong to syndetic, idempotent filters. Idempotent filters are combinatorial objects that abound in ergodic Ramsey theory but have been largely unnoticed and unexplored.

In order to investigate multiplicative structures in additively large sets, Beiglb ̈ock et al. raised a significant open question as to whether or not every subset of the natural numbers with bounded gaps (syndetic set) contains arbitrarily long geometric progressions.

2020年3月8日 · S ⊆ G is called a syndetic set if there is a finite set F ⊆ G such that G = FS. We can define respectively the upper density and lower density of S with respect to F by ¯ dF(S) = lim sup n → ∞ | S ∩ Fn | | Fn |, and d _ F(S) = lim inf n → ∞ | S ∩ Fn | | Fn |. Let S ⊆ G be a subset of G of positive upper density.

2022年12月12日 · A set $$A \subseteq {\mathbb {N}}$$ is called syndetic if there exists $$l \in {\mathbb {N}}$$ such that A intersects every set of l consecutive natural numbers. In 2006, Beiglböck et al. raised a question of whether a syndetic set …

2019年11月30日 · If $S$ is thickly syndetic, then the set of the numbers $i$ that start length-$n$ runs in $S$ must be syndetic, separately for every $n$. This means that for some $k$, in every length-$k$ interval of natural numbers you can find a position of a length-$n$ run in $S$.

1 天前 · A syndetic filter is a filter whose members are all syndetic. If (X, T) 𝑋 𝑇 (X,T) ( italic_X , italic_T ) is a minimal system, the filter in (1.4) is an example of a syndetic, idempotent filter. Thus, we see that every dynamically central syndetic set belongs to a syndetic, idempotent filter.

syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities ofA and B, there is a high density set of witnesses to the piecewise syndeticity of A + B. Most of the result are shown to hold more generally for subsets of Z d.Thekeytechnical tool is a Lebesgue density theorem for measure spaces induced by cuts