AI 中文总结
研究零和线性形式下集合\(A\)的密度限制，通过不同条件得出表示数与集合密度的关系，如特定\(\mathbf{b}\)形式及一般\(\mathbf{b}\)在间隙条件下的相关结论，恢复了关于\(B_{2k}\)序列的定理。
AI 中文摘要
设\(h \geq 2\)，\(\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h\)是坐标非零的零和向量。对于集合\(A=\{a_1\lt a_2\lt\cdots\}\subseteq\mathbb{N}\)，\(r_{A,\mathbf{b}}(n)\)表示\(A\)中两两不同元素的\(h\)元组\((x_1,\ldots,x_h)\)满足\(b_1x_1+\cdots+b_hx_h=n\)的个数。研究了使这些表示数保持较小的集合\(A\)的密度限制，得到了无限西顿集经典密度定理的类似结果。在\(\mathbf{b} = (c_1,-c_1,\dots,c_k,-c_k)\)的情况下，证明了若\(A(x)/(x/\log x)^{1/2k}\to\infty\)，则\(\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty\)；若\(A(x)\gg x^{1/2k}\)，则\(\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x\)，这恢复了陈关于\(B_{2k}\)序列的定理。对于一般零和向量\(\mathbf{b}\)，在间隙条件下证明了类似的界。
英文摘要
Let $h \geq 2$, and let $\mathbf{b} = (b_1,\dots,b_h)\in \mathbb{Z}^h$ be a zero-sum vector with nonzero coordinates. For a set $A=\{a_1<a_2<\cdots\}\subseteq\mathbb{N}$, let $r_{A,\mathbf{b}}(n)$ denote the number of $h$-tuples $(x_1,\ldots,x_h)$ of pairwise distinct elements of $A$ satisfying $b_1x_1+\cdots+b_hx_h=n$. We study density restrictions on sets $A$ for which these representation counts remain small, obtaining analogues of the classical density theorem for infinite Sidon sets. In the case $\mathbf{b} = (c_1,-c_1,\dots,c_k,-c_k)$, we prove that if $A(x)/(x/\log x)^{1/2k}\to\infty$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $A(x)\gg x^{1/2k}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$. This recovers Chen's theorem on $B_{2k}$-sequences. For general zero-sum vectors $\mathbf{b}$, we prove analogous bounds under gap conditions: if $a_{n+1}-a_n=o(n^{h-1}\log n)$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\to\infty$, whereas if $a_{n+1}-a_n\ll n^{h-1}$, then $\frac{1}{x}\sum_{|n|\leq x} r_{A,\mathbf{b}}(n)\gg\log x$.
Comments10 pages
Journal refMonatsh. Math. 211 (2026), no. 2, 315-327
DOI：10.1007/s00605-026-02211-4
