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I asked this question on math.SE before, but did get not get an answer. Therefore I hope it is ok to post it here on this site..

Let $q \in \mathbb{R}_+$ and $0 < w_1 \leq w_2 \leq \ldots w_d \in \mathbb{R}_+$. Define $$A_w(d,q):=\left\{{\bf k} \in \mathbb{N}_0^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ denote the number of non-negative integer points in the $\ell_1$-ellipse with semi-axes of length $\frac{q}{w_j}$ (which is a simplex). Similarly, $$B_w(d,q):=\left\{{\bf k} \in \mathbb{N}_+^d: \sum_{j=1}^d w_j k_j \leq q\right\}$$ denotes the number of positive integer points in this simplex.

My question is: Is there a relationship between $A_w(d,q)$ and $B_w(d,q)$ in the sense that if one could obtain an exact formula for $A_w(d,q)$ there also would be an exact formula for $B_w(d,q)$? For example something like $$B_w(d,q) = A_\tilde{w}(d,\tilde{q}) .$$

Thanks!

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  • $\begingroup$ Please provide a link to the question on math.SE. $\endgroup$
    – Stefan Kohl ♦
    Commented Aug 23, 2015 at 12:52
  • $\begingroup$ What are your assumptions on the $w_i?$ $\endgroup$
    – Igor Rivin
    Commented Aug 23, 2015 at 15:33
  • $\begingroup$ they are positive real numbers. I edited the question. $\endgroup$
    – User133713
    Commented Aug 23, 2015 at 16:29
  • $\begingroup$ Crossposted on math.stackexchange.com/questions/1405826/… $\endgroup$
    – Tadashi
    Commented Aug 23, 2015 at 20:31

2 Answers 2

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This does not answer your question in the most general setting, but check out the paper by Ricky Diaz and Sinai Robins.In particular, the Erhart-McDonald reciprocity to which they allude seems to be closely related to your question.

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I found a solution to the problem in this paper -- formulas (1.2) and (1.3)

http://homepages.math.uic.edu/~yau/35%20publications/An%20upper.pdf

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