I am working on a problem these days and the following issue came up. I am not sure yet 
that I understand it's depth very well, so I would like to discuss a simple case. For those 
interested, the problem has applications in coding theory. 


Consider a quadratic polynomial $f \left( x\_1, x\_2, x\_3, x\_4 \right)$ with real roots and 
coefficients drawn from a continuous distribution (and therefore irrationals with probability 1). 
Is there a strictly positive lower bound on $|f \left( x\_1, x\_2, x\_3, x\_4 \right)|$ if we constrain 
all $x\_1, x\_2, x\_3, x\_4$ to lie in $\mathbb{Z}$ ? In other words, is there a $\gamma > 0$ such 
that $\displaystyle |f \left( x\_1, x\_2, x\_3, x\_4 \right)| \geq \gamma~~ \text{for all} ~~ x\_1, x\_2, x\_3, x\_4 \in \mathbb{Z}$ ?


It seems to me that this is a Diophantine approximation - type problem. Note that one can 
show through a simple application of Khintchine-Groshev theorem that $|f \left( x\_1, x\_2, x\_3, x\_4 \right)|$ will be strictly positive for all $x\_1, x\_2, x\_3, x\_4 \in \mathbb{Z}$, in that case with probability 1. This is relatively straightforward. 



Diophantine approximations and quadratic polynomials
I think the answer is no. For example, the Oppenheim conjecture (proved by Margulis in 1987) states that if an indefinite nondegenerate quadratic form has at least 3 variables and it is not proportional to a rational quadratic form, then its set of values taken at integers are dense in the real line.


