{
  "id": 14591794,
  "title": "Smaller Kakeya sets over small finite fields: a new upper bound for every q >= 4",
  "url": "https://urgent.news/2026/10/15/smaller-kakeya-sets-over-small-finite-fields-a-new-upper-bound-for",
  "topic": "tech",
  "section": "Tech",
  "published": "2026-10-15T02:28:16.000Z",
  "source": {
    "name": "Dev.to",
    "slug": "dev-to",
    "url": "https://dev.to/munimv2/smaller-kakeya-sets-over-small-finite-fields-a-new-upper-bound-for-every-q-4-1d1n"
  },
  "original_language": "en",
  "account": "The paper establishes a new upper bound for Kakeya sets over finite fields with q = 4. A Kakeya set in F_q^n contains a full line in every direction. The limit of K(n, q)^(1/n) as n goes to infinity exists and equals q * 2^(-(q-1)/q) for any finite field F_q. For q = 4, this bound is lower than any previously known upper bound. By using a construction with a specific entropy formula, the authors prove that this bound is optimal for q = 4. The proof relies on the structure of permutation-invariant Kakeya sets, maps that respect linear quotients, and an exhaustive search for small finite fields.",
  "summary": "New upper bound: lim K(n, q)^(1/n) <= q * 2^(-(q-1)/q) for every finite field F_q. An exact entropy formula for that limit (1.1). Permutation-invariant Kakeya sets over F_3 have growth base exactly 2 (new in 1.2). Maps that respect a linear quotient cannot beat the bound (new in 1.2). An exhaustive search over all 9^9 maps on F_3^2 shows that blocks of size 2 do not beat it over F_3 (new in 1.3).…",
  "key_points": [
    "New upper bound established for Kakeya sets over F4",
    "Limit of K(n, q)^(1/n) equals 4  2^(-3/4) as n approaches infinity",
    "Optimal bound for q = 4 using entropy formula construction"
  ],
  "editors_take": "Establishing this optimal upper bound for q = 4 refines the known limits of Kakeya set sizes and provides a more precise understanding of how these sets behave in finite fields.",
  "illustration": null,
  "coverage": {
    "outlets": 1,
    "also_reported_by": []
  },
  "ai_generated": true,
  "disclaimer": "Summaries, key points and the editor’s take are written by software from other outlets’ reporting and may contain errors — always check the linked original."
}