Consider the set of polynomials with real coefficients as a vector space with the following inner-product: $\langle f, g \rangle = \int_{a}^{b} f(x)g(x) dx$.
Hilbert showed, in a paper from 1894, that the norm (with respect to this inner-product) of a non-zero polynomial in $\mathbb{Z}[X]$ can get arbitrarily small when $b-a < 4$. In other words, $\min_{0 \neq p \in \mathbb{Z}[X]} \int_{a}^{b} p^2(x) dx = 0$.
My questions are:
What happens when $b-a \ge 4$? Can the norm get arbitrarily small in this case? Or is there some (positive) lower bound for $\min_{0 \neq p \in \mathbb{Z}[X]} \int_{a}^{b} p^2(x) dx$?
When $b-a < 4$, is there an explicit construction of a sequence $p_n \in \mathbb{Z}[X],n\ge 1$, with norm tending to 0?
I posted similar questions in MSE and got no responses.
(Hilbert's proof was as follows: Minimizing the norm for (non-zero) polynomials of degree less than $n$ is equivalent to minimizing a certain positive-definite quadratic form. The corresponding matrix $A_n$ has entries $a_{i,j} = \langle x^i, x^j \rangle = \frac{b^{i+j+1} - a^{i+j+1}}{i+j+1}, 0 \le i,j \le n-1$. A calculation using an orthonormal basis for our inner-product space shows that $\det A_n = (\frac{b-a}{4})^{n^2}n^{-1/4} (2 \pi)^n c_n$ where $c_n$ converges to a positive constant. A result by Minkowski shows that in general, the minimal value of a positive quadratic form $\langle v, Av \rangle$ in $n$ variables is at most $n (\det A)^{1/n}$. Since $\lim_{n} n (\det A_n)^{1/n} = 0$ for $b-a<4$, the result follows.)
