AI 中文总结
研究离散约翰型定理覆盖问题，通过证明在维度\(n\)中凸进展可被无限恰当广义算术进展覆盖，且其大小在原集合基数\(O(n)^{2n}\)倍内，改进了之前的\(O(n)^{3n}\)倍，同时指出\(\Omega(n)^n\)阶损失不可避免。
AI 中文摘要
陶哲轩和武向宇引入了约翰定理的离散类似物，其中凸进展由广义算术进展近似。在该问题的覆盖版本中，人们寻求一个包含给定原点对称凸体所有格点的小广义算术进展。我们证明，维度\(n\)中的每个这样的凸进展都允许一个无限恰当的广义算术进展覆盖，其大小在原集合基数的\(O(n)^{2n}\)倍以内，改进了之前已知的\(O(n)^{3n}\)倍。我们还表明，对于无限恰当的广义算术进展覆盖，\(\Omega(n)^n\)阶的损失是不可避免的。
英文摘要
Tao and Vu introduced a discrete analogue of John's theorem in which convex progressions are approximated by generalized arithmetic progressions. In the covering version of this problem, one asks for a small GAP containing all lattice points of a given origin-symmetric convex body. We prove that every such convex progression in dimension $n$ admits an infinitely proper GAP cover whose size is within a factor $O(n)^{2n}$ of the cardinality of the original set, improving the previously known factor $O(n)^{3n}$. We also show that a loss of order $Ω(n)^n$ is unavoidable for infinitely proper GAP covers.
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