Larger few-distance sets in ℝ⁶, ℝ⁷, ℝ⁸ and ℝ⁹ from shells of root lattices

A finite set X ⊂ ℝᵈ is an s-distance set if exactly s distinct distances occur between its points. Szöllősi and Östergård (Electron. J. Combin. 27 (2020), P1.23) tabulated the largest known s-distance sets in ℝᵈ for d ≤ 8 and s ≤ 6; for s ∈ {5, 6} and d ∈ {6, 7, 8} the entries come from truncated simplices, and the authors remark that these are "rather weak". We show that the vectors of norm at most 6 of the root lattices E₆, E₇, E₈ contain much larger few-distance sets whose squared distances are 2, 4, …, 2s. Subsets found by a maximum-clique local search have 133 and 227 points in ℝ⁶ (s = 5, 6), 130, 253 and 388 points in ℝ⁷ (s = 4, 5, 6) and 258, 520 and 910 points in ℝ⁸ (s = 4, 5, 6), against the previously recorded 112, 140; 127, 168, 280; and 241, 252, 504. The same method gives 4- and 5-distance sets of 235 and 440 points in ℝ⁹, where the best explicit constructions we are aware of have 210 and 360 points. For s = 5 the sets have the transparent form {0} ∪ R ∪ S, where R is the root system and S is a subset of the second shell with pairwise inner products at least −1; we prove a general lemma of this kind for every even lattice of minimal norm 2. All sets are given explicitly as integer vectors and are verified by an exact, dependency-free script. The authors are listed in alphabetical order and contributed equally.