An answer to Goswami's question and new sources of $IP^{\star}$-sets containing combined zigzag structure
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| Format: | Preprint |
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2024
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| _version_ | 1866913357623197696 |
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| author | Debnath, Pintu |
| author_facet | Debnath, Pintu |
| contents | $A$ set is called $IP$-set in a semigroup $\left(S,\cdot \right)$ if it contains finite products of a sequence. A set that intersects with all $IP$-sets is called $IP^\star$-set. It is a well known and established result by Bergelson and Hindman that if $A$ is an $IP^{\star}$-set, then for any sequence $\langle x_{n}\rangle_{n=1}^{\infty}$, there exists a sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ such that $FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\cup FP\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\subset A$. In \cite[Question 3]{G}, S. Goswami posed the question: if we replace the single sequence by $l$-sequences, then is it possible to obtain a sum subsystem such that all of its zigzag finite sums and products will be in $A$. Goswami has given affirmative answers only for dynamical $IP^{\star}$-sets which are not equivalent to those of $IP^{\star}$-sets, but are rather significantly stronger. In this article, we will give the answer to Goswami's question that was unknown until now. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_11433 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An answer to Goswami's question and new sources of $IP^{\star}$-sets containing combined zigzag structure Debnath, Pintu Combinatorics 05D10, 22A15, 54D35 $A$ set is called $IP$-set in a semigroup $\left(S,\cdot \right)$ if it contains finite products of a sequence. A set that intersects with all $IP$-sets is called $IP^\star$-set. It is a well known and established result by Bergelson and Hindman that if $A$ is an $IP^{\star}$-set, then for any sequence $\langle x_{n}\rangle_{n=1}^{\infty}$, there exists a sum subsystem $\langle y_{n}\rangle_{n=1}^{\infty}$ such that $FS\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\cup FP\left(\langle y_{n}\rangle_{n=1}^{\infty}\right)\subset A$. In \cite[Question 3]{G}, S. Goswami posed the question: if we replace the single sequence by $l$-sequences, then is it possible to obtain a sum subsystem such that all of its zigzag finite sums and products will be in $A$. Goswami has given affirmative answers only for dynamical $IP^{\star}$-sets which are not equivalent to those of $IP^{\star}$-sets, but are rather significantly stronger. In this article, we will give the answer to Goswami's question that was unknown until now. |
| title | An answer to Goswami's question and new sources of $IP^{\star}$-sets containing combined zigzag structure |
| topic | Combinatorics 05D10, 22A15, 54D35 |
| url | https://arxiv.org/abs/2405.11433 |