An improved lower bound for sets with no cube differences

Let D₃(N) be the largest size of a subset of {1, …, N} no two of whose elements differ by a nonzero cube. Ruzsa's construction gives D₃(N) ≫ N^γ(m) with γ(m) = (1/3)(2 + log α₃(m)/log m) for every squarefree m, where α₃(m) is the largest size of a subset of ℤ/mℤ with no nonzero cube difference. The best exponent previously recorded, γ(91) = (1/3)(2 + log 14/log 91) = 0.861681…, comes from a 14-element set modulo 91 = 7 · 13 found by Lewko. We exhibit an explicit 116-element set modulo 3367 = 7 · 13 · 37 with no nonzero cube difference, which gives D₃(N) ≫ N^γ with γ = (1/3)(2 + log 116/log 3367) = 0.861763…. The modulus was located by a structural analysis of the modular problem: only primes ≡ 1 (mod 3) can occur, the "base m³" variant of the construction gives nothing new, and a strong-product ceiling α₃(7m′) ≤ ⌊7α₃(m′)/2⌋ together with exact values of α₃(p) for all primes p ≤ 163 excludes every modulus 7p with p ≤ 163 and singles out 3367 as the natural candidate. The certificate was found by a tabu search over the fibres of the decomposition P₃(ℤ/3367ℤ) ≅ C₇ ⊠ P₃(ℤ/481ℤ); every finite claim is verified by an accompanying standard-library script.