  The level-$k$ $\ell_1$-Fourier weight of a Boolean function refers to the sum
of absolute values of its level-$k$ Fourier coefficients. Fourier growth refers
to the growth of these weights as $k$ grows. It has been extensively studied
for various computational models, and bounds on the Fourier growth, even for
the first few levels, have proven useful in learning theory, circuit lower
bounds, pseudorandomness, and quantum-classical separations.
  We investigate the Fourier growth of certain functions that naturally arise
from communication protocols for XOR functions (partial functions evaluated on
the bitwise XOR of the inputs to Alice and Bob). If a protocol $\mathcal C$
computes an XOR function, then $\mathcal C(x,y)$ is a function of the parity
$x\oplus y$. This motivates us to analyze the XOR-fiber of $\mathcal C$,
defined as $h(z):=\mathbb E_{x,y}[\mathcal C(x,y)|x\oplus y=z]$.
  We present improved Fourier growth bounds for the XOR-fibers of protocols
that communicate $d$ bits. For the first level, we show a tight $O(\sqrt d)$
bound and obtain a new coin theorem, as well as an alternative proof for the
tight randomized communication lower bound for Gap-Hamming. For the second
level, we show an $d^{3/2}\cdot\mathrm{polylog}(n)$ bound, which improves the
previous $O(d^2)$ bound by Girish, Raz, and Tal (ITCS 2021) and implies a
polynomial improvement on the randomized communication lower bound for the
XOR-lift of Forrelation, extending its quantum-classical gap.
  Our analysis is based on a new way of adaptively partitioning a relatively
large set in Gaussian space to control its moments in all directions. We
achieve this via martingale arguments and allowing protocols to transmit real
values. We also show a connection between Fourier growth and lifting theorems
with constant-sized gadgets as a potential approach to prove optimal bounds for
the second level and beyond.
