  We apply, for the first time, the Transit Least Squares (TLS) algorithm to
search for new transiting exoplanets. TLS is a successor to the Box Least
Squares (BLS) algorithm, which has served as a standard tool for the detection
of periodic transits. In this proof-of-concept paper, we demonstrate how TLS
finds small planets that have previously been missed. We showcase TLS'
capabilities using the K2 EVEREST-detrended light curve of the star K2-32
(EPIC205071984) that was known to have three transiting planets. TLS detects
these known Neptune-sized planets K2-32b, d, and c in an iterative search and
finds an additional transit signal with a high signal detection efficiency
(SDE_TLS) of 26.1 at a period of 4.34882 (-0.00075, +0.00069) d. We show that
this signal remains detectable (SDE_TLS = 13.2) with TLS in the K2SFF light
curve of K2-32, which includes a less optimal detrending of the systematic
trends. The signal is below common detection thresholds, however, if searched
with BLS in the K2SFF light curve (SDE_BLS = 8.9) as in previous searches.
Markov Chain Monte Carlo sampling shows that the radius of this candidate is
1.01 (-0.09, +0.10) Earth radii. We analyze its phase-folded transit light
curve using the vespa software and calculate a false positive probability FPP =
3.1e-3, formally validating K2-32e as a planet. Taking into account the
multiplicity boost of the system, FPP < 3.1e-4. K2-32 now hosts at least four
planets that are very close to a 1:2:5:7 mean motion resonance chain. The
offset of the orbital periods of K2-32e and b from a 1:2 mean motion resonance
is in very good agreement with the sample of transiting multi-planet systems
from Kepler, lending further credence to the planetary nature of K2-32e. We
expect that TLS can find many more transits of Earth-sized and smaller planets
in the Kepler data that have hitherto remained undetected with BLS and similar
algorithms.
