{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nIn 2024, Stefan Steinerberger proposed the quantitative Ryser conjecture, a strengthened form of the circulant Hadamard conjecture. It asserts that there is an absolute constant $\\varepsilon_0\u003e0$ such that, for every $n\u003e4$ and every Littlewood polynomial\n\n$$P(z)=\\sum_{k=0}^{n-1}a_kz^k,\\qquad a_k\\in\\{-1,1\\},$$\n\none has\n\n$$\\max_{0\\leq j\u003cn} \\left| \\left|P\\!\\left(e^{2\\pi i j/n}\\right)\\right|-\\sqrt n \\right| \\geq\\varepsilon_0 n^{1/4}.$$\n\nSteinerberger called this formulation *ultra-flat at roots of unity*.\n\nIf $A=\\operatorname{circ}(a_0,\\ldots,a_{n-1})$, its singular values are the numbers $|P(e^{2\\pi i j/n})|$. Thus the conjecture is equivalently the spectral-gap assertion\n\n$$\\max_{1\\leq j\\leq n} \\left|\\sigma_j(A)-\\sqrt n\\right| \\geq\\varepsilon_0n^{1/4},$$\n\nwhere $\\sigma_1(A),\\ldots,\\sigma_n(A)$ are the singular values of $A$. The ordinary Ryser conjecture says only that the left-hand side is nonzero; the quantitative version predicts a universal gap of order $n^{1/4}$.\n\nThe exponent $1/4$ would be optimal up to logarithmic factors. When $n$ is prime, probabilistic modifications of the [[Legendre symbol|01J8AMV9RS]] give circulant sign matrices satisfying\n\n$$\\max_{0\\leq j\u003cn} \\left| \\left|P\\!\\left(e^{2\\pi i j/n}\\right)\\right|-\\sqrt n \\right| =O\\!\\left(n^{1/4}\\sqrt{\\log n}\\right).$$\n\nConsequently, the power of $n$ in the conjectured lower bound cannot in general be increased beyond $1/4$.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "inventory": [
      "2RGTKVQ7XP"
    ],
    "kind": "statement",
    "name": "Quantitative Ryser conjecture",
    "role": "proposition",
    "sources": [
      {
        "locator": "Ryser's conjecture on circulant Hadamard matrices, section Quantitative Ryser conjecture",
        "mention": "wikipedia://en/82859189#G14P",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:82859189@r1369844038"
      }
    ],
    "tag": "ZSJCMR4A04"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/ZSJCMR4A04/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/ZSJCMR4A04.md",
  "msc": null,
  "name": "Quantitative Ryser conjecture",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZSJ/ZSJCMR4A04.md",
  "status": null,
  "tag": "ZSJCMR4A04",
  "uri": "m2://ZSJCMR4A04",
  "url": "https://math2.tamnd.com/wiki/quantitative-ryser-conjecture/",
  "writing": {
    "tag": "ZSJCMR4A04",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}