Inverse residues and Pyatetski-Shapiro sequences

Let $c>1$ be fixed non-integer. Then the set $\mathcal{P}_{c}$ of integers $m = \bigl[n^{c}\bigr]$, $n = 1,2,3,\ldots$ is called as Pyatetskii -Shapiro sequence. There are a lot of papers devoted to different number -theoretical problems with the elements of the sequences $\mathcal{P}_{c}$.
In the talk, we will speak about the distribution of inverse residues modulo $q$ for the elements of Pyatetskii -Shapiro sequence, that is, about the distribution of the solution of the congruence
$$ mm^{*}\,\equiv\,1 \pmod q $$
with the conditions $m\in \mathcal{P}_{c}$, $1\leqslant m\leqslant X$, where $X = X(c,q)\to +\infty$ as $q\to +\infty$.