Wikipedia:Reference desk/Archives/Mathematics/2025 April 10

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April 10

Kakeya Set

Hi. I have some questions about Kakeya Sets, just for fun.

1. Can a Kakeya set exist, given the additional restriction that it contain exactly one line segment for each angle? (Beyond the obvious answer of a circle)
2. Can such a set be made arbitrarily large? (my intuition was yes picturing a spiral shape, but I have a gut feeling that my intuition is wrong. )

Duomillia (talk) 00:32, 10 April 2025 (UTC)

I have an intuitive gut feeling that intuitions and gut feelings are the same thing. -- Jack of Oz ^{[pleasantries]} 00:57, 10 April 2025 (UTC)

My intuition tells me that we must not indulge in gut feelings.  ​‑‑Lambiam 11:39, 10 April 2025 (UTC)

My gut tells me not to eat ice cream after pizza. —Tamfang (talk) 18:29, 10 April 2025 (UTC)

Isn't the closure of the deltoid shown as the first illustration in our article Kakeya set an example meeting your additional restriction? Also, isn't every superset of a Kakeya set, up to $\mathbb {R} ^{2},$ also a Kakeya set?  ​‑‑Lambiam 11:47, 10 April 2025 (UTC)

To clarify, for 1. when you say "exactly one line segment for each angle", do you mean line segments centered at the origin? The example of the circle you gave contains multiple line segments for any particular orientation centered at any given point within the circle. GalacticShoe (talk) 17:02, 10 April 2025 (UTC)

If you think of the line segment as being oriented, if it makes an angle $\alpha$ with the horizontal (oriented from $-\infty$ to $+\infty$), turning it around by half a turn changes the angle to $\alpha +\pi$ (modulo $2\pi$). So when making a full turn in a disk of diameter $1$ it attains each angle $0\leq \alpha <2\pi$ precisely once.  ​‑‑Lambiam 17:26, 10 April 2025 (UTC)

Ah I see where I went off the rails, I failed to notice the "unit" part of "unit line segment" in the definition of a Kakeya set, in which case yeah the unit circle would clearly work. I imagine the Reuleaux triangle, or any Reuleaux polygon, would be another example. GalacticShoe (talk) 17:35, 10 April 2025 (UTC)

Reduced Dedekind psi function?

The Euler's totient function is:

$\varphi (n)={\begin{cases}1&{\text{if }}n=1\\(p-1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\\varphi (a)\varphi (b)&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

and we have the “reduced” Euler's totient function: (sequence A011773 in the OEIS)

$\lambda (n)={\begin{cases}1&{\text{if }}n=1\\(p-1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\lcm(\lambda (a),\lambda (b))&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

Also, the Dedekind psi function is:

$\psi (n)={\begin{cases}1&{\text{if }}n=1\\(p+1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\\psi (a)\psi (b)&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

and we have the “reduced” Dedekind psi function:

$\gamma (n)={\begin{cases}1&{\text{if }}n=1\\(p+1)p^{r-1}&{\text{if }}n=p^{r}{\text{ with }}p{\text{ prime}}\\lcm(\gamma (a),\gamma (b))&{\text{if }}n=ab{\text{ with }}gcd(a,b)=1\\\end{cases}}$

the values of the “reduced” Dedekind psi function $\gamma (1)$ to $\gamma (24)$ are 1, 3, 4, 6, 6, 12, 8, 12, 12, 6, 12, 12, 14, 24, 12, 24, 18, 12, 20, 6, 8, 12, 24, 12, but this sequence does not in OEIS, thus does this function exist in number theory? 220.132.216.52 (talk) 02:54, 10 April 2025 (UTC)

If the sequence is not in the OEIS that's usually a good indication that it's not well known. If it were it would be much more likely to be found there than here. --RDBury (talk) 13:53, 15 April 2025 (UTC)