math.CO — Combinatorics
A composite modulus and Frobenius digits in the Heilbronn triangle construction
OpenAI proved that for every sufficiently large n there are n points in the unit square all of whose triangles have area at least c n^(-2+eta), with an explicit but extremely small eta (of order 10^(-235)). Subsequent public work lowered the cost of the construction, the latest claimed exponent being eta = 2/100866719, about 1.98*10^(-8). We introduce two changes. First, the digit base is taken to be a multiple of the label prime r and the main modulus is h = r B^(k-1) instead of B^k; this reduces the ratio h/tau between the modulus and the excluded determinant window from about B to about r, which allows the extension degree d = 3. Second, the norm polynomial is placed in the digits by a digit-wise convolution along the Frobenius expansion of the exponents, with a trace digit in a polynomial basis, giving k = 1440 positions. The result is Delta(n) >= c n^(-2+eta) with eta = 1/177022 > 5.64*10^(-6), for every sufficiently large n. The finite identities are checked by exact computation in the ancillary files.