

Title: A linear algebra upper bound. Post by acarchau on Sep 14^{th}, 2009, 9:29pm Let X and Y be linearly independent vectors in R^2. Let the lattice U be the set of all vectors of the form: mX+nY, where m and n are integers. Choose an appropriately small and positive d, and let W(d) be the non empty set { v in R^2 : v + u > d for all u in U}. For any v in W(d) let g(v) = sup_{ u in U} (  u  /  u + v). Then is sup_{v in W(d)} g(v) < infinity? 

Title: Re: A linear algebra upper bound. Post by Eigenray on Sep 14^{th}, 2009, 10:41pm Each g(v) is finite but rather than sup_{v in W(d)} g(v) being finite, we have in fact g(v) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif as v http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/to.gif http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif. Indeed, there exists a constant C such that for all v, there exists u in U with u+v<C. For v > R, pick such a u; then C > u+v http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ge.gif v  u > R  u, so u > R  C, and g(v) http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/ge.gif u/u+v > (RC)/C, which goes to infinity as R does. 

Title: Re: A linear algebra upper bound. Post by acarchau on Sep 15^{th}, 2009, 5:44pm Nice argument. Thanks. 

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