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 $Ax^2 + By^2 + Cz^2 \leq Z$?


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 $M.$).

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  • $\begingroup$ @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

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