Covering a Connected Curve on the Torus with Squares |
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Authors: | Shang-Yuan Shiu |
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Institution: | 1. Institute of Mathematics, Academia Sinica, Taipei City, 10617, Taiwan 2. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112-0090, USA
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Abstract: | We throw i.i.d. random squares S 1,S 2,… with respective side lengths l 1,l 2,… uniformly on the two-dimensional torus ?/?×?/?, where $\{l_{n}\}_{n=1}^{\infty}$ is a nonincreasing sequence with 0<l n <1 and lim n→∞ l n =0. A necessary and sufficient condition for covering the connected curve {0}×?/? is $$\sum_{n=1}^{\infty}\frac{l_n}{(\sum_{i=1}^{n}l_i)^2}\exp{\Biggl(\sum _{i=1}^{n}l_i^2\Biggr)}=\infty.$$ |
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