{
  "id": "2006.03558",
  "version": "v1",
  "published": "2020-06-05T17:11:31.000Z",
  "updated": "2020-06-05T17:11:31.000Z",
  "title": "Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications",
  "authors": [
    "Vitaly Bergelson",
    "Joel Moreira",
    "Florian K. Richter"
  ],
  "comment": "38 pages",
  "categories": [
    "math.DS",
    "math.CO"
  ],
  "abstract": "We obtain new results pertaining to convergence and recurrence of multiple ergodic averages along functions from a Hardy field. Among other things, we confirm some of the conjectures posed by Frantzikinakis in [Fra10; Fra16] and obtain combinatorial applications which contain, as rather special cases, several previously known (polynomial and non-polynomial) extensions of Szemeredi's theorem on arithmetic progressions [BL96; BLL08; FW09; Fra10; BMR17]. One of the novel features of our results, which is not present in previous work, is that they allow for a mixture of polynomials and non-polynomial functions. As an illustration, assume $f_i(t)=a_{i,1}t^{c_{i,1}}+\\cdots+a_{i,d}t^{c_{i,d}}$ for $c_{i,j}>0$ and $a_{i,j}\\in\\mathbb{R}$. Then $\\bullet$ for any measure preserving system $(X,{\\mathcal B},\\mu,T)$ and $h_1,\\dots,h_k\\in L^\\infty(X)$, the limit $$\\lim_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^N T^{[f_1(n)]}h_1\\cdots T^{[f_k(n)]}h_k$$ exists in $L^2$; $\\bullet$ for any $E\\subset \\mathbb{N}$ with $\\overline{\\mathrm{d}}(E)>0$ there are $a,n\\in\\mathbb{N}$ such that $\\{a,\\, a+[f_1(n)],\\ldots,a+[f_k(n)]\\}\\subset E$. We also show that if $f_1,\\dots,f_k$ belong to a Hardy field, have polynomial growth, and are such that no linear combination of them is a polynomial, then for any measure preserving system $(X,{\\mathcal B},\\mu,T)$ and any $A\\in{\\mathcal B}$, $$\\limsup_{N\\to\\infty}\\frac{1}{N}\\sum_{n=1}^N\\mu\\Big(A\\cap T^{-[ f_1(n) ]}A\\cap\\ldots\\cap T^{-[f_k(n)]}A\\Big)\\,\\geq\\,\\mu(A)^{k+1}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-05T17:11:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple ergodic averages",
      "hardy field",
      "combinatorial applications",
      "recurrence",
      "convergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}