  We obtain upper bounds on the cardinality of Hilbert cubes in finite fields,
which avoid large product sets and reciprocals of sum sets. In particular, our
results replace recent estimates of N. Hegyv\'ari and P. P. Pach (2020), which
appear to be void for all admissible parameters. Our approach is different from
that of N. Hegyv\'ari and P. P. Pach and is based on some well-known bounds of
double character and exponential sums over arbitrary sets, due to A. A.
Karatsuba (1991) and N. G. Moshchevitin (2007), respectively.
