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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.

john mangual
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  • An additional natural question is to have a reasonable upper bound on $m\mapsto\inf{\max(x,y,z):x,y,z$ such that $0<|q(x,y,z)|<1/m}$. – YCor Jan 09 '17 at 18:00
  • I had some success reading the article Values of quadratic forms at primitive integral points by Dani and Margulis, which proves a slightly stronger version of Oppenheim, and in particular implies that the values of the specific quadratic form in the questions are dense. As to @Ycor's question, these notes by Dani have a section on 'Quantitative Oppenheim', and a Google search for that phrase leads to a plethora of results. Unfortunately I don't know what the state of the art is. – John Griesmer Jan 09 '17 at 18:13
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    I'll just repeat what I can learn from Wikipedia https://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 https://terrytao.wordpress.com/2007/09/29/ratners-theorems/ . – David E Speyer Jan 09 '17 at 19:54
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    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. – Kevin Buzzard Jan 09 '17 at 20:04
  • Sorry, forgot to say, quadratic form in at least three variables, and indefinite should be taken to include nondegenerate. – David E Speyer Jan 09 '17 at 21:01
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    John, in this precise case, Wikipedia provides reliable references, freely accessible online, e.g. Borel's survey: http://www.ams.org/journals/bull/1995-32-02/S0273-0979-1995-00587-2/ – YCor Jan 09 '17 at 22:20
  • Just to add some references, the exact problem (or more specifically, search bound for this problem) is tackled in a paper by Lindenstrauss and Margulis - http://www.ma.huji.ac.il/~elon/Publications/oppenheim.pdf , the general case of quantitative density of unipotent flows will appear in upcoming article of Margulis,Lindenstrauss,Mohammadi and Shah (and will be substantially more involved, as it quantifies the Dani-Margulis argument which was mentioned). – Asaf Feb 14 '17 at 22:57
  • @Asaf i never understood why we work so hard, when x,y,z can be produced fairly easily on the computer. Lindenstrauss is obviously correct, but you can even see my question got closed b/c I phrased it wrong – john mangual Feb 14 '17 at 23:25
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    @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. – Asaf Feb 15 '17 at 04:31
  • Anyhow, the fact that you can solve such inequality (the proof using measure-rigidity) does not give you any bound towards running a computer search or so (unlike what you will get for rational forms), that's why the Lindenstrauss-Margulis theorem is important, the fact that you can solve such questions in ubiquitous manner. Sometimes you can take shortucts, Elon likes to give the examples of $x^2+y^2+z^2+w^2-\alpha r^2$ or $x^2+y^2+z^2-\alpha r^2$ as examples for shortcuts, but the ternary case is generally open (apart from his theorem) for the best of my knowledge. – Asaf Feb 15 '17 at 04:34

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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.

Kevin Buzzard
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    Had the question only been phrased correctly, one could retort, per Bombieri, "I asked for non-trivial solutions." (Ignoring the correction in http://mathoverflow.net/questions/27305/a-binomial-generalization-of-the-flt-bombieris-napkin-problem#comment58579_27305 ….) Also, one might mention $(x, y, z) = (0, 0, 0)$. – LSpice Jan 09 '17 at 21:27