We present a survey of results related to the solution of Kolmogorov–Nikolsky problem for Fourier sums on the classes of generalized Poisson integrals C^α,r_β,p, which consists in finding of asymptotic equalities for exact upper boundaries o f uniform norms of deviations of partial Fourier sums on the classes of 2π–periodic functions C^α,r_β,p, which are defined as convolutions of the functions, which belong to the unit balls pf the spaces L_p, 1≤ p≤∞, with generalized Poisson kernels P_α,r,β(t)=∑_k=1^∞e^-α k^rcos(kt-βπ/2), α>0, r>0, β∈ℝ.
Let $A$ be a countable and discrete subset of ${\Bbb R}^d$, $d \ge 2$, of positive upper Beurling density. Let $K$ denote a bounded symmetric convex set with a smooth boundary and everywhere non-vanishing Gaussian curvature. It is known that ${\mathcal E}(A)=\{e^{2 \pi i x \cdot a}\}_{a \in A}$ cannot serve as an orthogonal basis for $L^2(K)$ \cite{IKT01}. In this paper, we prove that even approximate average orthogonality is an obstacle to the existence of an exponential frame in the following sense. Let $A$ be as above and $\phi \ge 0$ be a continuous monotonically nonincreasing function on $[0, \infty)$ such that the approximate orthogonality condition holds \begin{align}\notag {\left( \frac{1}{2^j} \int_{2^j}^{2^{j+1}} \phi^p(t) dt \right)}^{1/p} \leq c_j 2^{-j\frac{d+1}{2}} \quad \text{and} \quad |\widehat{\chi}_K(a-a')| \leq \phi(\rho^*(a-a')) \ \forall a \not=a , a,a' \in A, \end{align} where $\rho^*$ is the Minkowski functional on $K^*$, the dual body of $K$. Then, if $$\limsup_{j \to \infty} c_j=0,$$ then the upper density of $A$ is equal to $0$, hence ${\mathcal E}(A)$ is not a frame for $L^2(K)$. The case $p=\infty$ was previously established by the authors of this paper in \cite{IM2020}. The point is that if ${\mathcal E}(A)$ is a frame for $L^2(K)$, then very few pairs of distinct exponentials $e^{2 \pi i x.a}, e^{2 \pi i x.a'}$ from ${\mathcal E}(A)$ come anywhere near being orthogonal. Our proof uses a generalization of Bourgain's result on pinned distances determined by sets of positive Lebesgue upper density in ${\Bbb R}^d$, $d \ge 2$. We also improve the $L^{\infty}$ version of this result originally established in \cite{IM2020}. By using an extension of the combinatorial idea from \cite{IR03}, we prove that under the $L^{\infty}$ hypothesis, $A$ is finite if $d \not=1 \mod 4$. If $d=1$ mod $4, A$ may be infinite, but if it is, then it must be a subset of a line.
