发表机构
Mathematical Institute of the Serbian Academy of Sciences and Arts; Department of Mathematics, University of Kentucky; Yau Mathematical Sciences Center, Tsinghua University; Beijing Institute of Mathematical Sciences and Applications; Department of Mathematics, Stanford University（塞尔维亚科学与艺术学院数学研究所; 肯塔基大学数学系; 清华大学丘成桐数学科学中心; 北京国际数学研究中心; 斯坦福大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究线性方程整数解数量的渐近最大值，提出一般结构结果，并确定特定系数下的精确极限值。
AI 中文摘要
我们研究当$|S|\to +\infty$时，对于固定的$a, b, c \in \mathbb{Z}$以及有限集$S\subset \mathbb{Z}$中的变量$x, y, z \in S$，线性方程$ax+by+cz = 0$的整数解数量的渐近最大可能值。定义$\gamma_{a, b, c}$为最大的常数，使得存在任意大的有限集$S\subset \mathbb{Z}$，满足方程$ax+by+cz=0$在$x,y,z\in S$中的解的数量为$\gamma_{a,b,c}|S|^2-o(|S|^2)$。我们证明了一般$a, b, c$的结构性结果，并且进一步表明，对于某个常数$\delta>0$，有$5/13\le \gamma_{1,1,-3}\le 1/2-\delta$。此外，我们证明当$a \rightarrow \infty$时，$\gamma_{1,1,-a}$的极限恰好等于$1/5$。
英文摘要
We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $γ_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $γ_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le γ_{1,1,-3}\le 1/2-δ$ for some constant $δ>0$. In addition we show that the limit as $a \rightarrow \infty$ of $γ_{1,1,-a}$ is equal to precisely $1/5$.
Comments31 pages
