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does Oppenheim conjecture hold for specific quadratic forms? or for generic quadratic forms with a set of measure 1.

for example can we find $x,y,z \in \mathbb{Z}$ with $$|x^2 + y^2 - \sqrt{3} z^2| < 10^{-6}$$ is that implied by Oppenheim conjecture?

where are elementary expositions of proof? what is the current standing?? I know the proof involves homogeneous flows

is Wikipedia correct here? I thought it was almost all indefinite ternary quadratic forms $ax^2 + by^2 - cz^2$ with $[ a:b:c]$ not all rational.

Wikipedia is known to have dubious statements. here I am wondering what orbit(s) were used with Ratner theorem or any elementary proof that $10^{-100}$ is possible.

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  • $\begingroup$ An additional natural question is to have a reasonable upper bound on $m\mapsto\inf\{\max(x,y,z):x,y,z$ such that $0<|q(x,y,z)|<1/m\}$. $\endgroup$
    – YCor
    Commented Jan 9, 2017 at 18:00
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    $\begingroup$ I'll just repeat what I can learn from Wikipedia en.wikipedia.org/wiki/Oppenheim_conjecture . The Oppenheim conjecture states that if $Q$ is an indefinite quadratic form which is not a real multiple of an integer quadratic form, than $Q(\mathbb{Z}^n)$ takes values in $(0, \epsilon)$ for any $\epsilon>0$. So yes, this implies such $(x,y,z)$ exist. Margulis proved Oppenheim's conjecture in 1987. Terry Tao has a post on the deduction of Oppenheim from Ratner's orbit closure theorem terrytao.wordpress.com/2007/09/29/ratners-theorems . $\endgroup$
    – David E Speyer
    Commented Jan 9, 2017 at 19:54
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    $\begingroup$ I see. So the actual question is whether Oppenheim's conjecture holds for all quadratic forms or just most of them, and assuming Wikipedia is accurate this answers the question immediately. $\endgroup$
    – Kevin Buzzard
    Commented Jan 9, 2017 at 20:04
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    $\begingroup$ John, in this precise case, Wikipedia provides reliable references, freely accessible online, e.g. Borel's survey: ams.org/journals/bull/1995-32-02/S0273-0979-1995-00587-2 $\endgroup$
    – YCor
    Commented Jan 9, 2017 at 22:20
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    $\begingroup$ @johnmangual , first thing you misunderstood Oppenheim's conjecture, so I believe this is the reason why your question got closed (btw, if you're interested in a.e. result, you should check the recent work by Bourgain on the subject). But even if you believe the Oppenheim conjecture (and you should, this is a theorem by Margulis now), the easy proof (by Ratner's theorem) does not give you some effective manner how to find such a solution, Margulis' first proof is essentially effective, and his second proof (with Dani) can be made entirely effective, as was done in the article I've linked to. $\endgroup$
    – Asaf
    Commented Feb 15, 2017 at 4:31

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$x=6627,y=314048,z=238678$ is an answer to the question in the title but I'm sure that the real question is something I don't really understand.

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    $\begingroup$ Had the question only been phrased correctly, one could retort, per Bombieri, "I asked for non-trivial solutions." (Ignoring the correction in mathoverflow.net/questions/27305/… ….) Also, one might mention $(x, y, z) = (0, 0, 0)$. $\endgroup$
    – LSpice
    Commented Jan 9, 2017 at 21:27

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