  The entropy/influence conjecture, raised by Friedgut and Kalai in 1996, seeks
to relate two different measures of concentration of the Fourier coefficients
of a Boolean function. Roughly saying, it claims that if the Fourier spectrum
is "smeared out", then the Fourier coefficients are concentrated on "high"
levels. In this note we generalize the conjecture to biased product measures on
the discrete cube, and prove a variant of the conjecture for functions with an
extremely low Fourier weight on the "high" levels.
