Convergence exponent of singular integral in generalized Hilbert–Kamke problem

In this article we find exact value of the convergence exponent of singular integral in the problem of simultaneous representation of increasing set of natural numbers $N_1,\ldots,N_r$ by sum of terms $[x^{n_1+\theta}],[x^{n_2+\theta}],\ldots,[x^{n_r+\theta}]$ ($n_1<n_2<\ldots<n_r$ — natural numbers, $0\leq\theta\leq1$). We consider integral:
$$ \theta_0=\int\limits_{\mathbb R^r}|I(\alpha_1,\ldots,\alpha_r)|^k\,d\alpha_1\ldots d\alpha_r, $$
where $k$ is an unrestricted index and
$$ I(\alpha_1,\ldots,\alpha _r)=\int\limits_{0}^{1}\exp\biggl\{2\pi i\sum_{j=1}^{r}\alpha_jx^{n_j+\theta}\biggr\}\,dx. $$
It is proved that $\theta_0$ converges when $k>k_0$ and diverges when $k\leq k_0$ where
$$ k_0=\max \left\{n_1+\cdots+n_r+r\theta,\frac{r(r+1)}{2}+1\right\}. $$