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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 differentiable but $f’ \neq g’$?

Comments:

It is natural to ask if $f' \neq g'$ can hold even at a single point. The following example shows that this can indeed hold. Let $r_n \to 0$ be a slowly decreasing sequence of positive numbers, and set

$$f(x) = \sum_n r_n \mathbf 1_{[r_{n+1}, r_n)} (x),$$ $$ g(x) = 2f(x).$$

Then $f' = g' = 0$ a.e., but $f$ and $g$ are additionally differentiable at $0$ with derivatives $1$ and $2$ respectively.

Replacing the piecewise constant staircase with a Cantor staircase, we can even arrange for $f$ and $g$ to be continuous.

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 differentiable but $f’ \neq g’$?

Comments:

It is natural to ask if $f' \neq g'$ can hold even at a single point. The following example shows that this can indeed hold. Let $r_n \to 0$ be a slowly decreasing sequence of positive numbers, and set

$$f(x) = \sum_n r_{n+1} \mathbf 1_{[r_{n+1}, r_n)} (x),$$ $$ g(x) = 2f(x).$$

Then $f' = g' = 0$ a.e., but $f$ and $g$ are additionally differentiable at $0$ with derivatives $1$ and $2$ respectively.

Replacing the piecewise constant staircase with a Cantor staircase, we can even arrange for $f$ and $g$ to be continuous.

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 differentiable but $f’ \neq g’$?

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 differentiable but $f’ \neq g’$?

Comments:

It is natural to ask if $f' \neq g'$ can hold even at a single point. The following example shows that this can indeed hold. Let $r_n \to 0$ be a slowly decreasing sequence of positive numbers, and set

$$f(x) = \sum_n r_n \mathbf 1_{[r_{n+1}, r_n)} (x),$$ $$ g(x) = 2f(x).$$

Then $f' = g' = 0$ a.e., but $f$ and $g$ are additionally differentiable at $0$ with derivatives $1$ and $2$ respectively.

Replacing the piecewise constant staircase with a Cantor staircase, we can even arrange for $f$ and $g$ to be continuous.

Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the maximal Hausdorff dimension $d$ (and corresponding Hausdorff $d$-measure) of the set on which $f$ and $g$ are both differentiable but $f’ \neq g’$?

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 differentiable but $f’ \neq g’$?