A quantitative version of a de Bruijn-Post theorem (Q2762663)

The theorem by de Bruijn and Post mentioned in the title says that if \(f\) on \([0,1]\) is not Riemann integrable then there exists a uniformly distributed sequence \(x_n\) such that the sequence \(\frac 1n \sum_{i=1}^n f(x_i)\) does not converge. NEWLINENEWLINENEWLINEThe authors now consider a bounded function \(f:[0,1]\to {\mathbb R}^d\) and denote by \(\Lambda\) the set of all possible limit points of Riemann sums \(S_n(f,T) = \sum_{i=1}^n f(t_i)(x_i-x_{i-1})\) (where \(0=x_0< x_1 < \cdots < x_n = 1\) is a partition of \([0,1]\), \(t_i\in [x_i,x_{i-1}]\) and \(\max_i |x_i-x_{i-1}|\to \) as \(n\to\infty\)). It is known that \(\Lambda\) is non-empty, compact and convex. Their main result (Theorem 3.8) says that for every connected closed set \(C\subseteq \Lambda\) there exists a uniformly distributed sequence \(x_n\) such that the set of all limits of convergent subsequences of NEWLINE\[NEWLINE \frac 1n \sum_{i=1}^n f(x_i) NEWLINE\]NEWLINE coincides with \(C\).