  We present a simple proof on the existence of $L^1$-flat analytic polynomials
with coefficients $0,1$ on the circle and on the real line and we give an
example of a conservative ergodic map and flow whose unitary operators admits a
simple Lebesgue spectrum. Among other results, we obtain an answer to
Bourgain's question on the supremum of $L^1$-norm of such polynomials and to a
question inspired by Lehmer's problem on the supremum of the Mahler measures of
those polynomials.
