Polynomials with integer coefficients, with value close to $0$, in the interval $[-1,1]$ Are there some interesting properties of polynomials with integer coefficients of degree $2^n$ which satisfy $\mid P(x) \mid \le \frac{1}{2^k}$ ?
I know that their coefficients are bounded and the bound is $4e^d$ where $d$ is the degree. As the coefficients are bounded, the roots are also bounded.
The extremal version of the problem is "Integer Chebyshev" problem.
I also have some more properties like $P(0) = 0$, $P(1) = 0$, $P(-1) = 0$, and any $k$ roots of the polynomial can be known, where $k$ is bounded by polynomial of $n$, but the degree is $2^n$.
Are there more properties about distribution of their zeroes, how their derivatives behave, how these polynomials behave outside $[-1,1]$ ?
 A: It is too long for a comment so I will post it as an answer. After
some discussion in the comments the question had been reformulated as follows:
What are the "interesting" properties of the polynomial with integer coefficients such that $\|P_n\|_{C[-1,1]}\le 1.$
Since Chebyshev polynomial satisfies this condition and is an extremal polynomial for many max/min problems in approximation theory it is quite hard to summarize everything. For a good account of such problems see http://books.google.ca/books/about/Analytic_Theory_of_Polynomials.html?id=FzFEEVO3PXYC&redir_esc=y
and http://books.google.ca/books?id=386CC7JnuuwC&printsec=frontcover&dq=Borwein+Erdelyi&hl=en&sa=X&ei=ez2LUbLKCI34rAHS6ID4CA&ved=0CDIQ6AEwAA
Since you mentioned behaviour of derivative, I will give one example here:
Markov's inequality: $\|P_n\|_{C[-1,1]}\le 1$ implies $\|P_n'\|_{C[-1,1]}\le n^2.$  One can generalize this result to higher derivatives.
bubba's observation is also correct: if $0<k<1$ then at least one of the coefficients of $kT_n(x)$ is non integer. Indeed, if this is not the case, $k\cdot 2^n$ has to be integer and therefore $k=\frac{p}{2^m}$ $m\ge 1.$
But using recurrence relation $T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)$ and induction it is easy to show that at least of the coefficients of $T_n$ is odd. This provides a contradiction.
