Divergence almost everywhere of rectangular partial sums of multiple Fourier series of bounded functions

In this paper we establish the following results, which are the multidimensional generalizations of well-known theorems:

• 1) Suppose that a function $f\in C(\mathbb T^m)$ has no intervals of constancy in $\mathbb T^m$; then there exists a homeomorphism $\varphi\colon\mathbb T^m\to\mathbb T^m$ such that the Fourier series of the superposition $F=f\circ\varphi$ is divergent with respect to rectangles almost everywhere;
• 2) for any integrable function $f\in L^1(\mathbb T^m)$, with $|f(\mathbf x)|\geqslant\alpha>0$, $x\in\mathbb T^m$, there exists a signum function $\varepsilon(\mathbf x)=\pm 1$, $\mathbf x\in\mathbb T^m$ such that the Fourier series of the product $f(\mathbf x)\varepsilon(\mathbf x)$ is divergent with respect to rectangles almost everywhere.