On the Maximum Number of Translates in a Point Set
Given a finite set P subset of R(d), called a pattern, t(P)(n) denotes the maximum number of translated copies of P determined by n points in R(d). 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 R(d) are rational simplices. The function t(P)(n) is also determined exactly when vertical bar P vertical bar <= 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 subset of R(+) determined by n positive reals. As a consequence, we show that s(Ak)(n) = n - Theta(n(1-1/pi(k))), where A(k) = {1, 2,..., k} is an arithmetic progression of size k, and pi(k) is the number of primes less than or equal to k.
translates pattern translated copies rational simplex scaled copies geometric pattern kruskal-macaulay rational dimension