Yu 2008 note b 2 g sets

../

---

Gang Yu, A note on B_2[g] sets. Integers: Electronic Journal of Combinatorial Number Theory 8 (2008), #A58.

Let F(g,N) be the largest size of a B_2[g] set in {1,...,N}, meaning every integer has at most g representations as an unordered sum of two elements. Green's bound F(g,N) <= sqrt(1.75(2g-1)N) had been the best for small g, giving about sqrt(5.25N) at g = 2. Theorem 1.1 improves the constant to 1.74246, so F(2,N) <= sqrt(5.2274N), about 2.2864 sqrt(N), and Theorem 1.2, using a different weight function (Section 4), lowers it to 1.74217. The method combines Green's fourth-moment estimate for the Fourier transform of a B_2[g] set (Lemma 2.1, from the author's earlier paper) with a new choice of even test weight w on [-1,1] satisfying two sign conditions, yielding the bound F(g,N) <= sqrt(c_w N) with c_w an explicit functional of w (Lemma 2.2); Yu notes the constant is not the limit of the method. For problem 158 the paper is a direct finite B_2[g] source, refining the weighted fourth-moment approach; it has since been beaten numerically by Habsieger-Plagne and by White, and it gives no infinite nesting or control on the liminf of the normalized counting function.

Source: https://www.integers-ejcnt.org/vol8.html. The file prints no license line; the journal's own site (https://www.integers-ejcnt.org) could not be read on 2026-10-02, and its volume page at https://math.colgate.edu/~integers/vol8.html (read 2026-10-02) lists the article under DOI 10.5281/zenodo.10131097, whose Zenodo record of the journal's deposit states the license "Creative Commons Attribution 4.0 International" (read 2026-10-02), the Creative Commons Attribution 4.0 license.

Bears on. #158

Results to transcribe.

• Theorem 1.1: For every g >= 2, F(g,N) <= (1+o(1))sqrt(1.74246(2g-1)N); in particular F(2,N) <= (1+o(1))sqrt(5.2274N).
• Theorem 1.2: With an improved weight function, F(g,N) <= (1+o(1))sqrt(1.74217(2g-1)N) for every g >= 2.
• Lemma 2.1: Fourth-moment estimate: for any fixed eps in (0,1/2), the sum of |f(n/2N)|^4 over 1 <= n <= N^eps is at most (2g-1)N|A|^2 - |A|^4/2, up to (1+o(1)).
• Lemma 2.2: For an even weight w with continuous second derivative and the two stated sign conditions, F(g,N) <= (1+o(1))sqrt(c_w N) with c_w an explicit functional of w and g.