四元多重和集的局部分类
A Local Classification of Four-Element Multiple Sumsets
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中文总结 AI 辅助
研究有限集\(A\subset\mathbb{Z}\)的\(h\)重和集相关问题，核心方法是利用Lev的下界简化问题到仿射直径为五的归一化集，主要贡献是确定\(R(h,4)\)特定区间的值，证明相关猜想并得到新缺失区间。
中文摘要 AI 辅助
对于有限集\(A\subset\mathbb{Z}\)，用\(hA\)表示其\(h\)重和集，记\(R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}\)。本文确定了\(R(h,4)\)中位于\(4h + 2\)和\(6h - 4\)之间的部分：\(h = 4\)时唯一值是\(5h - 1\)，\(h\geq5\)时唯一值是\(5h - 1\)和\(5h + 1\)。证明了Rajagopal猜想的\(5h\notin R(h,4)\)对\(h\geq4\)成立，\(h\geq6\)时还得到新的缺失区间\([5h + 2,6h - 4]\)。通过Lev的下界将问题简化为仿射直径为五的归一化集，经反射和四个基本精确和集计算完成分类。
英文摘要
For a finite set $A\subset\mathbb{Z}$, write $hA$ for its $h$-fold sumset, and let \[ R(h,k)=\{|hA|:A\subset\mathbb{Z},\ |A|=k\}. \] We determine the part of $R(h,4)$ lying between $4h+2$ and $6h-4$: for $h=4$ the only value is $5h-1$, while for $h\geq 5$ the only values are $5h-1$ and $5h+1$. This proves Rajagopal's conjectured gap $5h\notin R(h,4)$ for every $h\geq 4$. For $h\geq 6$, it also yields the new missing interval $[5h+2,6h-4]$, which lies outside Rajagopal's general excluded set. Lev's lower bound for the successive growth of multiple sumsets reduces the problem to normalized sets of affine diameter five, of which there are only six. Reflection and four elementary exact sumset computations finish the classification.
