We improve the best known upper bound for the bracketing number of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$). More precisely, we provide a better upper bound for the cardinality of an algorithmic bracketing cover construction due to Eric Thi\'emard, which forms the core of his algorithm to approximate the star discrepancy of arbitrary point sets from [E. Thi\'emard, An algorithm to compute bounds for the star discrepancy, J.~Complexity 17 (2001), 850 -- 880]. Moreover, the new upper bound for the bracketing number of anchored axis-parallel boxes yields an improved upper bound for the bracketing number of arbitrary axis-parallel boxes in $[0,1]^d$. In our upper bounds all constants are fully explicit.

翻译：我们改进了$d$维原点锚定轴平行盒（或等价地，与$d$维单位立方体$[0,1]^d$相交的左下正交区域）的括号覆盖数的最佳已知上界。具体而言，针对Eric Thiémard在构造算法化括号覆盖时提出的基数上界（该构造是其算法框架的核心，用于逼近任意点集的星偏差，源自E. Thiémard，《计算星偏差上界的算法》，J. Complexity 17 (2001), 850–880），我们给出了更优的上界。此外，锚定轴平行盒括号覆盖数的新上界进一步改进了$[0,1]^d$中任意轴平行盒的括号覆盖数上界。本文给出的所有上界均包含完全显式的常数。