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Let $A$ be a (Borel-)measurable subset of $[0,1]$.Let $\lambda$ denote Lebesgue measure. Is it possible that there exists a constant $c>0$ such that for all intervals $I \subset [0,1]$ we have $$ \lam …

I am a number theorist by profession, so apologies if the answer to this question is "trivially true" or "trivially false". Let $(\Omega, \mathcal{A}, P)$ be a (non-atomic) probability space. Let $(\ …

By Koksma's inequality, $s_n$ is bounded by $N$ times the variation of $f$, multiplied with the so-called discrepancy $D_N$ of the sequence $(\alpha, 2\alpha, \dots, N \alpha)$. The discrepancy of cou …

Let $f(x_1, \dots, x_n)$ be a real function on the $n$-dimensional unit cube (that is, mapping $[0,1]^n \mapsto \mathbb{R}$). Assume furthermore that $f$ is monotonic in every coordinate, and that $f$ …

The solution should be $b-a$. I don't know how difficult this is to prove from scratch, but I think it follows for example from work of W.M. Schmidt, see: Schmidt, Wolfgang M., A metrical theorem in g …