An experiment to measure single-spin asymmetries in semi-inclusive production of charged pions in deep-inelastic scattering on a transversely polarized $^3$He target was performed at Jefferson Lab in the kinematic region of $0.16<x<0.35$ and $1.4<Q^2<2.7$ ${rm GeV^2}$. The pretzelosity asymmetries on $^3$He, which can be expressed as the convolution of the $h^perp_{1T}$ transverse momentum dependent distribution functions and the Collins fragmentation functions in the leading order, were measured for the first time. Using the effective polarization approximation, we extracted the corresponding neutron asymmetries from the measured $^3$He asymmetries and cross-section ratios between the proton and $^3$He. Our results show that for both $pi^{pm}$ on $^3$He and on the neutron the pretzelosity asymmetries are consistent with zero within experimental uncertainties.
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