Let $h(x,y)$ be a polynomial with real coefficients. Suppose there are infinitely many integer solutions to $|h(x,y)|<1$. What can I say about $h$?

When $h$ itself has integer coefficients, a famous theorem of Siegel tells me that the curve $h(x,y)$ has geometric genus zero and either $1$ or $2$ points at infinity. The main reason I can make no progress on this question is that I know no analogous result for $h \in \mathbb{R}[x,y]$.

All I need is something very crude. Basically, if $(x_n,y_n)$ is the sequence of solutions ordered by $x_n^2+y_n^2$, and you can give me any reasonable upper bound on the growth rate of $x_n^2+y_n^2$, that is good enough to solve the linked problem. (When $h$ has integer coefficients, it follows from Siegel's theorem that $(x_n, y_n)$ are more or less the images of $(a_n, b_n)$ under a polynomial map, where $(a_n, b_n)$ are the solutions to a Pell equation, or else are of the form $(f(n), g(n))$ for some polynomials $f$ and $g$. So $x_n^2+y_n^2$ can grow at worst exponentially.)

Adding more details here: If $h$ has integer coefficients, then $|h(x,y)| < 1$ is the same as $h(x,y)=0$. Curves of genus $\geq 2$ have only finitely many rational points (Faltings). Affine curves of genus $1$ have only finitely many integer points (Siegel). $\mathbb{P}^1 \setminus \{ 0,1, \infty \}$ has only finitely many $\mathcal{O}_{K,S}$ points for any number field $K$ and any finite $S$ (this is the $S$-unit equation, I think finiteness was also proved by Siegel.) If the normalization of our curve is isomorphic over $\bar{\mathbb{Q}}$ to $\mathbb{A}^1 \setminus \{ z_1, z_2, \ldots, z_s \}$ for some $s \geq 2$ and some $z_i \in \bar{\mathbb{Q}}$ then, after extending the ground field and inverting finitely many primes, we can apply a linear change of variables making $z_1=0$ and $z_2=1$. So we can embed integer solutions into the $S$-unit equation for some $(K,S)$.

Thus, the only remaining options are a genus $0$ curve with one puncture or two punctures.

A genus zero curve with one puncture is rational over $\mathbb{Q}$, since it has a point (the puncture). So there is a parametrization $(f(t), g(t))$ of $h(x,y)=0$ where $f$ and $g$ are polynomials with rational coefficients. I haven't quite been careful with the details here, but the integer points should wind up being the image of some finite collection of arithmetic progressions under $(f(t), g(t))$.

A genus zero curve with two punctures is either $uv=1$ or $u^2-D v^2 = C$ for some nonsquare $D$. Once again, we get a parametrization $(u,v) \to (f(u,v), g(u,v))$ for some polynomials $(f,g)$ with rational coefficients. Again, we need to be careful with denominators from the coefficients of $(f,g)$, but we should get more or less the image of a pell sequence under a polynomial map in the second case, and only finitely many integer points in the first case.

I'm sure that I have seen (Faltings)+(Siegel)+($S$-unit) all stated together as "only finitely many integer points on a curve with $3g+n \geq 3$" and treated as Siegel's result; but I couldn't quickly find a reference that puts it that way.

The Markoff and Lagrange Spectra,where real coefficient indefinite binary quadratic forms are discussed. $\endgroup$ – Will Jagy Aug 7 '13 at 20:43