Let $L=[a,b]\cap\mathbb{N}$ with $a,b\in\mathbb{N}$, let $D\in\mathbb{N}$, and let $C=L^D$. Then I would like to know how many points are there in $C$ with the same given norm-2 $d$. I.e., I'm looking for $|A|$, where

$A = \{p\in C,\ ||p||_2=d\}$

  • 1
    $\begingroup$ What kind of answer do you want? There won't be a simple nice closed formula. But for example, you might fix $a,b,d$ and ask for an estimate for the size of $|A|$ as $D\to\infty$. Or you might fix $D$ and let $a,b,d\to\infty$ in a suitable way. Of course, there are trivial cases, for example, if $a>d/\sqrt{D}$, then $A=\emptyset$. $\endgroup$ – Joe Silverman Oct 28 '15 at 21:24
  • $\begingroup$ @JoeSilverman a nice-formula lower boundary will be the best option... I was thinking on using this for data compression but know I think it's not possible to apply it where I want. Having a lower limit will help me out proving that it is indeed stupid to keep researching on this path. $\endgroup$ – Carlos Navarro Astiasarán Oct 30 '15 at 0:28

The answer is the coefficient of $t^{d^2}$ in the generating function $\left( \sum_{j=a}^b t^{j^2}\right)^D$.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.