  Aims. We analyze the population of periodic variable stars in the open
cluster NGC 3766 based on a 7-year multi-band monitoring campaign conducted on
the 1.2 m Swiss Euler telescope at La Silla, Chili.
  Methods. The data reduction, light curve cleaning and period search
procedures, combined with the long observation time line, allow us to detect
variability amplitudes down to the milli-magnitude level. The variability
properties are complemented with the positions in the color-magnitude and
color-color diagrams to classify periodic variable stars into distinct
variability types.
  Results. We find a large population (36 stars) of new variable stars between
the red edge of slowly pulsating B (SPB) stars and the blue edge of delta Sct
stars, a region in the Hertzsprung-Russell (HR) diagram where no pulsation is
predicted to occur based on standard stellar models. The bulk of their periods
ranges from 0.1 to 0.7 d, with amplitudes between 1 and 4 mmag for the majority
of them. About 20% of stars in that region of the HR diagram are found to be
variable, but the number of members of this new group is expected to be higher,
with amplitudes below our milli-magnitude detection limit.
  The properties of this new group of variable stars are summarized, and
arguments set forth in favor of a pulsation origin of the variability, with
g-modes sustained by stellar rotation. Potential members of this new class of
low-amplitude periodic (most probably pulsating) A and late-B variables in the
literature are discussed.
  We additionally identify 16 eclipsing binary, 13 SPB, 14 delta Sct and 12
gamma Dor candidates, as well as 72 fainter periodic variables. All are new
discoveries.
  Conclusions. We encourage to search for the existence of this new class of
variables in other young open clusters, especially in those hosting a rich
population of Be stars.
