On conditions of boundedness of polynomials on a segment in the case when they are uniformly bounded on some subset of this segment (Q1857408)

Let \(P_{n}\left( x\right) \) be a real-valued polynomial, defined on \(\left[-1,1\right] \), which is uniformly bounded on \(E\subset\left[ -1,1\right] \). The reviewed article is devoted to solve the problem: under what conditions on \(E\) it may be concluded that \(P_{n}\left( x\right) \) is uniformly bounded on the whole segment \(\left[ -1,1\right] \)? For its solution in the artile a characteristic function \(\rho\left( E\right) =\max\{\inf\{\left|x-y\right|:y\in E\}:x\in\left[ -1,1\right] \}\) is introduced. The main result is: if for a given \(\theta>0\) and a natural \(n\geq 17\theta\) the condition \(\rho\left( E\right) \geq\theta n^{-2}\) is satisfied, then for every polynomial \(P_{n}\left( x\right) \) of degree \(n\) \[ \sup\{\left|P_{n}\left( x\right) \right|:x\in\left[ -1,1\right] \}/\sup\{\left|P_{n}\left( x\right) \right|:x\in E\}\leq c\left( \theta\right) , \] where \(c\left( \theta\right) \) does not depend on \(n\). It seems that the results of the paper may be helpful in solving the \textit{M. I. Kadec} problem on the distribution of points of the Tchebyshev alternance [Usp. Mat. Nauk 15, 199-202 (1960; Zbl 0136.36402)].