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This question resisted attempts on MSE. Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold: $$\li …

Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the supremal Hausdorff dimension of the set on which $f$ and $g$ are both different …

Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$. Let $\Omega …

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ …

This is a direct follow up to For which rationals is this exponential sum bounded? Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$. What is the Hausdorff dimension of the …

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$. That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$. Question: What is …

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure. What is the supremal Hausdorff dimension of the set on which $f$ is …