On divergence of Fourier–Walsh series of bounded functions on sets of measure zero

It is known that for an arbitrary number $p$, $1\leqslant p<\infty$, and any set of measure zero there exists a function in $L^p(0,\, 1)$ whose Fourier–Walsh–Paley series diverges on the set. In this paper we prove an analogous result in the case $p=\infty$ for Fourier–Walsh series (Fourier–Walsh–Paley series and Fourier–Walsh–Kaczmarz series).