Separated sumsets: upper bounds for B_{3,Δ}- and B_{4,Δ}-sets of real numbers

Abstract

Call a finite set S of real numbers a 1-separated B_h set if any two distinct h-element multisets from S have sums differing by at least 1. These are the real, unit-separation cases of the B_{h,Δ}-sets recently introduced by Nathanson. We prove that a 1-separated B₃ set S ⊆ [0,V] satisfies |S| ≤ (2/μ₂²)^{1/3} V^{1/3} (1+o(1)) < 1.51547 V^{1/3} as V → ∞, and that a 1-separated B₄ set satisfies |S| ≤ (4/μ₂²)^{1/4} V^{1/4} (1+o(1)) < 1.62431 V^{1/4}, where μ₂² is the L² autoconvolution constant. These are the first upper bounds in the h=3 and h=4 cases of Nathanson's extension problem for B_{h,Δ}-sets to improve on trivial packing-type counting, and the constants coincide with the best known bounds for ordinary integer B₃ and B₄ sets, due to White: for these two cases, at the current frontier of knowledge, the separated-real relaxation costs nothing asymptotically. The proof combines windowed counting lemmas, in which the separation hypothesis reproduces exactly the multiset rigidity available in the integer case, with a Fejér-kernel smoothing argument that feeds the autoconvolution constant directly, bypassing the variational machinery of Green.