  We focus on the fact that light-pulse atom interferometers measure the atoms'
acceleration with only three data points per drop. As a result, the measured
effect of the gravity gradient is systematically larger than the true one, an
error linear with the gradient and quadratic in time almost unnoticed so far.
We show how this error affects the absolute measurement of the gravitational
acceleration $g$ as well as ground and space experiments with gradiometers
based on atom interferometry such as those designed for space geodesy, the
measurement of the universal constant of gravity and the detection of
gravitational waves. When atom interferometers test the universality of free
fall and the weak equivalence principle by dropping different isotopes of the
same atom one laser interrogates both isotopes and the error reported here
cancels out. With atom clouds of different species and two lasers of different
frequencies the phase shifts measured by the interferometer differ by a large
amount even in absence of violation. Systematic errors, including common mode
accelerations coupled to the gravity gradient with the reported error, lead to
hard concurrent requirements --on the ground and in space-- on several
dimensionless parameters all of which must be smaller than the sought-for
violation signal.
