{
  "id": "1903.08731",
  "version": "v1",
  "published": "2019-03-20T20:40:30.000Z",
  "updated": "2019-03-20T20:40:30.000Z",
  "title": "Three Convolution Inequalities on the Real Line with Connections to Additive Combinatorics",
  "authors": [
    "Richard C. Barnard",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "math.CO"
  ],
  "abstract": "We discuss three convolution inequalities that are connected to additive combinatorics. Cloninger and the second author showed that for nonnegative $f \\in L^1(-1/4, 1/4)$, $$ \\max_{-1/2 \\leq t \\leq 1/2} \\int_{\\mathbb{R}}{f(t-x) f(x) dx} \\geq 1.28 \\left( \\int_{-1/4}^{1/4}{f(x) dx}\\right)^2$$ which is related to $g-$Sidon sets (1.28 cannot be replaced by 1.52). We prove a dual statement, related to difference bases, and show that for $f \\in L^1(\\mathbb{R})$, $$ \\min_{0 \\leq t \\leq 1}\\int_{\\mathbb{R}}{f(x) f(x+t) dx} \\leq 0.42 \\|f\\|_{L^1}^2,$$ where the constant 1/2 is trivial, 0.42 cannot be replaced by 0.37. This suggests a natural conjecture about the asymptotic structure of $g-$difference bases. Finally, we show for all functions $f \\in L^1(\\mathbb{R}) \\cap L^2(\\mathbb{R})$, $$ \\int_{-\\frac{1}{2}}^{\\frac{1}{2}}{ \\int_{\\mathbb{R}}{f(x) f(x+t) dx}dt} \\leq 0.91 \\|f\\|_{L^1}\\|f\\|_{L^2}$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-20T20:40:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "convolution inequalities",
      "additive combinatorics",
      "real line",
      "difference bases",
      "connections"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}