{
  "id": "2310.18873",
  "version": "v1",
  "published": "2023-10-29T02:15:51.000Z",
  "updated": "2023-10-29T02:15:51.000Z",
  "title": "Combined exponential patterns in multiplicative $IP^{\\star}$ sets",
  "authors": [
    "Pintu Debnath",
    "Sayan Goswami"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "$IP$ sets play fundamental role in arithmetic Ramsey theory. A set is called an additive $IP$ set if it is of the form $FS\\left(\\langle x_{n}\\rangle_{n\\in \\mathbb{N}}\\right)=\\left\\{ \\sum_{t\\in H}x_{t}:H\\right.$ is a nonempty finite subset of $\\left.\\mathbb{N}\\right\\}$, whereas it is called a multiplicative $IP$ set if it is of the form $FP\\left(\\langle x_{n}\\rangle_{n\\in \\mathbb{N}}\\right)=\\left\\{ \\prod_{t\\in H}x_{t}:H\\right.$ is a nonempty finite subset of $\\left. \\mathbb{N}\\right\\}$ for some injective sequence $\\langle x_{n}\\rangle_{n\\in \\mathbb{N}}.$ An additive $IP^{\\star}$ (resp. multiplicative $IP^{\\star}$) set is a set which intersects every additive $IP$ set (resp. multiplicative $IP$ set). In \\cite{key-1}, V. Bergelson and N. Hindman studied how rich additive $IP^{\\star}$ sets are. They proved additive $IP^{\\star}$ sets ($AIP^{\\star}$ in short) contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto in\\cite{key-3}, where he proved that multiplicative $IP^{\\star}$ sets ($MIP^{\\star}$ in short) contain exponential tower\\footnote{will be defined later} and finite product of a single sequence. However exponential patterns can be defined in two different ways. In this article we will prove that $MIP^{\\star}$ sets contain two different exponential patterns and finite product of a single sequence. This immediately improves the result of A. Sisto. We also construct a $MIP^\\star$ set, not arising from the recurrence of measurable dynamical systems. Throughout our work we will use the machinery of the algebra of the Stone-\\v{C}ech Compactification of $\\mathbb{N}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-29T02:15:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D10"
    ],
    "keywords": [
      "exponential patterns",
      "finite product",
      "single sequence",
      "nonempty finite subset",
      "multiplicative"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}