Abstract
Given a finite set P⊆ℝd, called a pattern, t P (n) denotes the maximum number of translated copies of P determined by n points in ℝd. We give the exact value of t P (n) when P is a rational simplex, that is, the points of P are rationally affinely independent. In this case, we prove that t P (n)=n−m r (n), where r is the rational affine dimension of P, and m r (n) is the r -Kruskal–Macaulay function. We note that almost all patterns in ℝd are rational simplices. The function t P (n) is also determined exactly when | P |≤3 or when P has rational affine dimension one and n is large enough. We establish the equivalence of finding t P (n) and the maximum number s R (n) of scaled copies of a suitable pattern R⊆ℝ+ determined by n positive reals. As a consequence, we show that \(s_{A_{k}}(n)=n-\varTheta (n^{1-1/\pi(k)})\) , where A k ={1,2,…,k} is an arithmetic progression of size k, and π(k) is the number of primes less than or equal to k.
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Research partially supported by PIFI 3.1–3.2, UAM-I-CA-53 Análisis Aplicado, when the first two authors visited B. Llano.
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Ábrego, B.M., Fernández-Merchant, S. & Llano, B. On the Maximum Number of Translates in a Point Set. Discrete Comput Geom 43, 1–20 (2010). https://doi.org/10.1007/s00454-008-9111-9
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DOI: https://doi.org/10.1007/s00454-008-9111-9