wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) riddles >> putnam exam (pure math) >> A linear algebra upper bound. (Message started by: acarchau on Sep 14th, 2009, 9:29pm)

Title: A linear algebra upper bound.
Post by acarchau on Sep 14th, 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 14th, 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| > (R-C)/C,
which goes to infinity as R does.

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