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arXiv · 2403.09585

A combinatorial approach to exponential patterns in multiplicative $IP^{\star}$ sets in $\mathbb{N}$

Pintu Debnath
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Abstract

In [On $IP^{\star}$sets and central sets, Combinatorica, 14 (1994) 269-277], N. Hindman and V.Bergelson proved additive $IP^{\star}$-sets contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto in [Exponential triples, Electronics Journal of Combinatorics, 18 (2011), no. 147], where he proved that multiplicative $IP^{\star}$-sets contain exponential $IP$ of type $I$ and finite sums of a single sequence as well as exponential $IP$ of type $II$ and finite products of another single sequence, using the algebra in the Stone-\v{C}ech Compactification of discrete semigroups. In this article, we will provide a combinatorial proof of the result of A. Sisto.

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Pintu Debnath. 2024-03-14. A combinatorial approach to exponential patterns in multiplicative $IP^{\star}$ sets in $\mathbb{N}$. https://arxiv.org/abs/2403.09585

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