Smaller Kakeya sets over small finite fields: a new upper bound for every q >= 4
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).…
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.
Written by urgent.news from Dev.to's reporting — not their text. Machine-written — may contain errors; check the original before relying on it.
