2
$\begingroup$

I have recently been studying some results about equidistribution of lattice points on positive definite quadratic forms $Q$ with $3$ or more variables. Concretely the article Duke and Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids (Inventiones, 1990) and the chapter 11 of Iwaniec Topic in classical automorphic forms.

These texts shows that by choosing appropriately the "radius" $\sqrt{n}$, we can ensure that the integer points of $n=Q(k_1,k_2,\ldots,k_d)$ are asymptotically equidistributed. What is more, we have infinitely many suchs $n$. My question is the following:

If I additionally impose the conditions that some of the variables $k_1,\ldots,k_d$ cannot be multiples of a finite set of primes, say $p_1,\ldots,p_r$; may I still find infinite $n$ with asymptotically equidistribution?

For example, in the easiest case. Is there a sequence $n_i$ tending to infinity such that we have asymptotically equidistribution of integer points as $n_i=Q(k_1,\ldots,k_d)$ with all the $k_1$ satisfying $p_1\nmid k_1$,$\ldots$,$p_r\nmid k_1$? What if we impose the same condition for $k_2$? And for $k_1$,$k_2$,$\ldots$?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ $k\not\mid p$ means $k$ doesn't divide $p$. Do you mean $p\not\mid k$? $\endgroup$
    – Gerry Myerson
    Commented Oct 11, 2022 at 4:45
  • 1
    $\begingroup$ Totally! Thank you. I have just corrected it. $\endgroup$
    – MathqA
    Commented Oct 11, 2022 at 9:51

0

Reset to default

You must log in to answer this question.