Three-dimensional topological insulator (TI) nanowires with quantized surface subband spectra are studied as a main component of Majorana bound states (MBS) devices. However, such wires are known to have large concentration $N sim 10^{19}$ cm$^{-3}$ of Coulomb impurities. It is believed that a MBS device can function only if the amplitude of long-range fluctuations of the random Coulomb potential $Gamma$ is smaller than the subband gap $Delta$. Here we calculate $Gamma$ for recently experimentally studied large-dielectric-constant (Bi$_{1-x}$Sb$_x$)$_2$Te$_{3}$ wires in a small-dielectric-constant environment (no superconductor). We show that provided by such a dielectric-constant contrast, the confinement of electric field of impurities within the wire allows more distant impurities to contribute into $Gamma$, leading to $Gamma sim 3Delta$. We also calculate a TI wire resistance as a function of the Fermi level and carrier concentration due to scattering on Coulomb and neutral impurities, and do not find observable discrete subband-spectrum related oscillations at $N gtrsim 10^{18}$ cm$^{-3}$.
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