Reciprocity (Ehrhart-style) for real polytopes? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T22:27:49Z http://mathoverflow.net/feeds/question/111747 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111747/reciprocity-ehrhart-style-for-real-polytopes Reciprocity (Ehrhart-style) for real polytopes? Sinai Robins 2012-11-07T19:21:24Z 2012-11-12T03:30:23Z <p>Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices? </p> <p>In other words, given any real closed (and convex) polytope $P$, is the function that enumerates integer points in dilates of $P$ somehow simply related to the function that enumerates integer points in the dilates of the interior of $P$?</p> <p>Here's one possible direction, given that we no longer have the usual quasi-polynomial behavior for the integer point enumerator. Does there exist some piecewise smooth function $L(t)$, of a real variable $t$, which satisfies the following properties:</p> <p>(1) $L(t)$ agrees with the integer point enumerator #{${\mathbb Z}^d \cap tP$} at all positive integer values of $t$.</p> <p>(2) $L(-t)$ counts the integer points in the interior of $tP$, for all positive integers $t$.</p> <p>(3) The Fourier transform of $L$ is supported at a countable set of points.</p> <p>Intuitively, for all real and positive $t$, such a smooth function $L(t)$ would approximate the piecewise constant function #{${\mathbb Z}^d \cap tP$}). Property (3) above was suggested by Allen Knutson.</p> <p>(ADDED) Let's try to see what happens in dimension one. So we let $P := [0, \alpha]$, a one-dimensional polytope, with $\alpha$ a positive irrational number. We let $L_P(n)$ be the number of integer points in the dilation $nP$, so that here $L_P(n) = [n\alpha]+1$. For the interior $P^o$, we have $L_{P^o}(n) = [n\alpha]$. Using {x} $= \frac{1}{2} + \sum_{k\in \mathbb Z - {0}} \frac{1}{k}e^{2\pi i k x}$, and $L_P(n)= n\alpha -$ {$n\alpha$}+1, we have: $$L_P(n) = [n\alpha]+1 = n\alpha + \frac{1}{2} - \sum_{k \in \mathbb Z - {0} } \frac{1}{k}e^{2\pi i k n \alpha},$$ and let's use this last expression as the extension of $L_P(n)$ to all real values of $n$. We now check for reciprocity, using this new definition of our extended function $L_P(n)$: $$L_P(-n) = -n\alpha + \frac{1}{2} - \sum_{k \in \mathbb Z - {0} } \frac{1}{k}e^{-2\pi i k n \alpha}$$ $$= -n\alpha + \frac{1}{2} + \sum_{k \in \mathbb Z - {0} } \frac{1}{k}e^{2\pi i k n \alpha} \ \ (\text{replacing } k \in \mathbb Z \ by \ -k \in \mathbb Z)$$ $$= - [ n\alpha]= - L_{P^o}(n),$$ which is a reciprocity law. </p>