[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"A4t0xrnl5a":3},"# Square-difference-free sets in F_3[T] past the conjectured bound\n\nLet `P_{q,n}` be the polynomials of degree less than `n` over the field with `q` elements, and call\na subset square-difference-free when no two of its elements differ by a nonzero square. For odd `q`\nand `4 ∣ n`, Naslund (*Paley graphs and Sárközy's theorem in function fields*, arXiv:2203.01293v3,\nQuart. J. Math. 74 (2023) 627–637, Theorem 1) constructs a square-difference-free subset of\n`P_{q,n}` with `q^{3n/4}` elements, and conjectures that none is larger. His Conjecture 13 reads:\n\"Let k ≥ 2, and suppose that gcd(k, q − 1) > 1. For n ≡ 0 (2k), any set A ⊂ P_{q,n} that does not\ncontain a k-th power difference has size at most |A| ≤ q^{n(1 − 1/k²)}. In particular, for k = 2\nand q odd, we conjecture that Theorem 1 is tight.\" This repository proves the case `k = 2`, `q = 3`\nfalse: the bound `3^{3n/4}` fails for every `n ≡ 0 (mod 4)` with `n ≥ 8`.\n\nThe six theorems:\n\n- a square-difference-free subset of `P_{3,8}` with 810 elements — `NS.D3_8_ge_810`;\n- 810 exceeds the conjectured `3^6 = 729`, so the bound fails at `n = 8` — `NS.conjecture13_fails_3_2_8`;\n- for every `e ≥ 1`, such a subset of `P_{3,8e}` with `810^e` elements — `NS.D3_8e_ge`;\n- for every `e`, such a subset of `P_{3,8e+4}` with `27 · 810^e` elements — `NS.D3_8e4_ge`;\n- the bound fails at every `n ≡ 0 (mod 4)` with `n ≥ 8` — `NS.conjecture13_fails_3_2_all`;\n- `16/21 ≤ liminf log D_3(n) / (n log 3)`, where `D_3(n)` is the largest size of a\n  square-difference-free subset of `P_{3,n}` — `NS.liminf_ge`.\n\nOne lift, applied repeatedly, gives all six. Fix the ten-word code\n`S = {0000, 0211, 0121, 0112, 1200, 1020, 1002, 2212, 2122, 2221} ⊂ F_3^4`, in which no two words\ndiffer by a vector with every coordinate in `{0, 1}`, the two squares of `F_3`. Put `P = T^3 - T`\nand `Q = P^2`. For even `m` and a square-difference-free `B ⊆ P_{3,m}`, the polynomials\n`V_s + P·R + Q·(b + s_∞ T^m + u T^{m+1})`, with `s ∈ S`, `R` of degree below 3, `u ∈ F_3` and\n`b ∈ B`, form a square-difference-free subset of `P_{3,m+8}` with `810 · |B|` elements. Iterating\nfrom the bases `{0} ⊆ P_{3,0}` and `B_4 = {aT^3 + bT + c(1 - T^2)} ⊆ P_{3,4}`, which has 27\nelements, reaches every `n` divisible by 4.\n\nNo upper bound on `D_3(n)` is proved and no value of `D_3(n)` is determined, at `n = 4` or anywhere\nelse; the ten-word code is not claimed to be optimal; nothing is claimed at any `q` other than 3.\nWrite `I_3(K)` for the largest size of a subset of `F_3^K` in which no two elements differ by a\nvector with every coordinate in `{0, 1}`; the code was found in the course of this work and shows\n`I_3(4) ≥ 10`. As of 2026-09-16 no published construction exceeds q^{3n/4} for\nsquare-difference-free subsets of F_q[T]; Naslund's Conjecture 13 has two indexed citing papers,\nneither addressing it; the small values I_3(3) = 4 and I_3(4) = 10 are stated as data whose prior\nappearance is under literature review (Calderbank–Frankl–Graham–Li–Shepp 1993 give the LP bounds\n5 and 11).\n\nRun locally with the pinned Lean and Mathlib versions; there are no GitHub Actions workflows.\n\n```sh\nlake build\npython scripts/check-source.py\npython scripts/check_construction.py\n```\n\n[PROOF.md](PROOF.md) gives the mathematics with the Lean name of every step,\n[VERIFICATION.md](VERIFICATION.md) the checks and their limits, and [DISCLOSURE.md](DISCLOSURE.md)\nthe assistance statement. [Challenge.lean](Challenge.lean) states the six theorems;\n[Solution.lean](Solution.lean) proves them.\n",1789847409414]