4
$\begingroup$

Let $A,B,C$ be positive integers, and let $Z$ be a positive parameter. Let $M$ be a positive integer, and consider the set of points

$$\displaystyle \{(x,y,z) \in \mathbb{Z}^3 : Ax^2 + By^2 + Cz^2 \leq MZ, Ax^2 + By^2 + Cz^2 \equiv 0 \pmod{M}\}.$$

Let $N(M;Z)$ denote the cardinality of the set above. How does $N(M;Z)$ compare to the cardinality of the set of integer points inside the ellipsoid $Ax^2 + By^2 + Cz^2 \leq Z$?

$\endgroup$
3
$\begingroup$

Bill Duke, On ternary quadratic forms, J. Number Theory 110 (2005), no. 1, 37--43, (preprint) gives a uniform estimate for the number of representations of a number by a ternary positive definite quadratic form, which implies the expected answer (that the points in the ellipsoid are equidistributed with respect to $M.$).

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ @ScottMorrison Thanks for the edit! $\endgroup$ – Igor Rivin Feb 24 '16 at 12:17
  • $\begingroup$ You're welcome. I've been trying out the new citation tool (the little "link" button in the edit toolbar), by occasionally inserting full citations into top-of-the-front-page posts. $\endgroup$ – Scott Morrison Feb 24 '16 at 22:25
  • $\begingroup$ @ScottMorrison Yes, I wondered how you did that, I will try to play with it... $\endgroup$ – Igor Rivin Feb 24 '16 at 23:25

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.