Bonus problem: find an arrangement of 4 queens on the board such that:

1. There is exactly 1 square on which a bishop can be placed, such that the 4 queens and the bishop attack all unoccupied squares

AND

2. There is exactly 1 square on which a rook can be placed, such that the 4 queens and the rook attack all unoccupied squares

Amazingly, modulo rotation / reflection, this problem has exactly 1 solution.

After identifying solutions up to rotation and reflection there are only 49 solutions. No solutions have rotational symmetry, and there is exactly one solution with reflection symmetry (already mentioned by an earlier commenter).

Out of the 49 solution classes, there are 18 distinct queen layouts. The layouts have between 1 and 5 ways to place the bishop to complete the solution. Interestingly, there is exactly one queen layout (up to rotation / reflection) for which there are exactly 2 ways to place the bishop to complete the puzzle.

Topological dimension is indeed something one can define: e.g. the Koch snowflake [1] or the graph of the Weierstrass function [2] have topological dimension 1. Actually, the first is homeomorphic to the unit circle and the second is homeomorphic to real line. It's great if you are doing topology and you only care about how things look like up to homeomorphism. But if you have metric structure (and you care about it), it is not so useful.

Minkowski dimension is certainly easy to define but it has some problems: sets which are "very small" (like a sequence `1/log(n)`) can have Minkowski dimension 1. The article has a minor technical oversight: the limit certainly does not need to exist. Minkowski defined it as the limit supremum of the sequence (actually, he defined it in terms of the decay rate of the size of the neighbourhood of the set, but this is equivalent). But one could analogously define a "lower" variant by taking the limit infimum instead.

Hausdorff dimension is not discussed in this article, but it is probably the most "robust" notion of dimension one can define. The Hausdorff dimension of any sequence is 0. But even then, lots of sets with Hausdorff dimension 1 can be very small, like the fat Cantor set which has dimension 1 but has length 0 [3]. So this 'dimension' does not necessarily line up with the intuition for "1-dimensional" in esoteric circumstances.

But even Hausdorff / Minkowski dimension does not capture the essence of some matters. For example, one might be interested in when a certain space can be mapped into another space without too much distortion (let's say by a map which respects the metric, like a bi-Lipschitz map). It can easily happen that a set has small (finite) Hausdorff or Minkowski dimension, but it cannot be embedded in a non-distorting way in any finite dimensional Euclidean space. This happens for instance with the real Heisenberg group [4]. If you are interested in this type problem then you want something like Assouad dimension [5].

The moral of the story is: the correct notion of dimension depends critically on what you want to do with your notion of 'dimension'. For sets which are very nice (smooth manifolds) all "reasonable" notions of dimension will coincide with what you expect; but beyond this there is an infinite zoo of ways to define dimension which are all reasonable in various ways, but capture genuinely different notions of 'size'.

[1]: https://en.wikipedia.org/wiki/Koch_snowflake

[2]: https://en.wikipedia.org/wiki/Weierstrass_function

[3]: https://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%9...

[4]: https://en.wikipedia.org/wiki/Heisenberg_group

[5]: https://en.wikipedia.org/wiki/Assouad_dimension

First, forget briefly about the Hilbert curve and just think about the unit square [0,1]^2.

If you take any point (x,y) in the unit square, we can associate x and y with binary coordinates, say x = 0.a1a2a3... and y = 0.b1b2b3... Then we can just define a new number z with binary representation 0.a1b1a2b2a3b3... And going the other way, given z in [0,1], we can take the 'even' binary coordinates to get x, and the 'odd' binary digits to get y.

The problem with this specific mapping is that the function is not continuous. But if you are a bit more careful:

1. The first digit says "left half vs right half"

2. The second digit says "top half vs bottom half" (of the rectangle from 1)

3. The third digit says "left half vs right half" (of the square from 2)

etc.

and then if two numbers share the first n binary digits (i.e. your points are close on the real line) then the corresponding points in the plane will also be quite close together (they are inside the same square / rectangle with side length like 2^(-n/2) at step n).

The "reason" why the dimension is different is precisely because of the "n/2": for every n digits of precision you have in the number z, you only get n/2 digits of precision for each of (x, y).

This is a bit imprecise because of issues with non-unique binary representation but (at least for me) it captures the spirit of why this should work!

I visited Josh at UBC last year around this time and I recall asking him if he thought Kakeya in dimension 3 would be solved soon. I remember that he believed it would be (though perhaps he was worried by someone other than Hong and himself). In the end they were able to complete the proof themselves.

Hong is probably quite a serious candidate for the fields medal because of this. She has already made impressive progress on problems in harmonic analysis and geometric measure theory and she is one of few people has a firm foothold in both fields at the same time.

The quanta article talks about a 'tower of conjectures' in harmonic analysis; at the top of the tower is the so-called "local smoothing conjecture" (a conjecture about how much waves, such as 'idealized' sound waves, can amplify from some initial configuration when averaged over time). A Kakeya set is a certain type of geometric obstruction to local smoothing; resolving the full conjecture also requires handling so-called 'oscillatory' obstructions. In dimension 2 + 1 (2 spatial and 1 time dimension) the local smoothing was only recently resolved (also by Hong and co-authors [3]); even though the corresponding result for Kakeya sets in dimension 2 has been known for over 40 years.

[1] https://player.vimeo.com/video/1062254156

[2] https://player.vimeo.com/video/1063428579

[3] https://annals.math.princeton.edu/2020/192-2/p06

For instance, this definition would not include objects such as the Devil's staircase [1] or more generally images of the unit interval under a continuous monotonically increasing function.

Some more exposition about attempts to rigorously define the notion of a fractal can be find in the introduction to Kenneth Falconer's book [2]

[1] https://en.wikipedia.org/wiki/Cantor_function

[2] https://zbmath.org/1285.28011

Edit: Fixed incorrect zbmath link.

I originally wrote this article for a “microcourse” I ran at the University of St Andrews—aimed at a non-tech background—on building a personal webpage. Especially in mathematics, having a personal site (that you control) to host research and other information is pretty invaluable!

Older personal math sites tend to have a very particular “historic” feel. While I personally have a lot of nostalgia for the look, I also think it’s good to take advantage of some of the newer tools that are available today!

One reason for believing that irrational algebraics, or pi, or e are normal, is a crude heuristic which is that "there is nothing special about base b". You can also take a computer and compute digit frequencies up to some very large precision, and see what happens. Generally speaking, it feels like "naturally defined numbers" should be normal unless there is a good reason for them not to be (and this is entirely independent of the fact that uniformly random numbers are normal). Proving this is a very different matter!

1) Birkhoff ergodic theorem, which states for a "nice" dynamical system, the probability that certain events occur can be described explicitly by an invariant distribution (see [1]), and

2) Continued fractions have an associated "nice" dynamical system (the Gauss map) which has an explicit probability distribution that is not too challenging to compute.

Of course, writing this argument out takes a bit of work [2].

In fact, the argument is structured in the exact same way as the fact that uniformly randomly chosen numbers in [0,1] are normal (i.e. the digit frequencies in a base-b expansion are all 1/b).

However, proving such results about _specific_ numbers is notoriously hard [3]. As far as I am aware, there has not been a single irrational algebraic number proven to be normal. Normality of well-known constants like pi and e is also an open problem! I would not be surprised if proving distributional results for continued fraction expansions of pi is also very hard.

[1]: https://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorem...

[2]: http://www.geometrie.tugraz.at/karpenkov/cf2011/cf2011s_7.pd...

[3]: https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...

A site to host my research, and some miscellaneous blog posts. I also have a recipes site:

https://food.rutar.org

They're all made using Zola, "compiled" to HTML + CSS.

The authors proved [1, Thm. 1.3] that given any two sets in R^d with equal non-zero measure and boundaries that are "not too horrible" (i.e. box / Minkowski dimensions of their boundaries less than d), one can cut one of the sets into finitely many Borel pieces and rearrange them (i.e. apply isometries in R^d) to obtain the other set.

You can also guarantee that the pieces have positive measure under a mild technical assumption.

[1] https://arxiv.org/pdf/2202.01412.pdf

A sketch of the argument in the IFS goes like this: consider the set of maps {f1, f2, f3} as acting on compact subsets of R^2 by the action F -> f1(F) U f2(F) U f3(F). Since the maps fi are continuous, the image is a compact set. But one can also prove that this action is a contraction mapping on the space of compact subsets of R^2 equipped with Hausdorff metric [1], and appealing to the Banach fixed point theorem [2],

(1) there is a unique compact set K fixed by this action, and

(2) the images of F converge "geometrically fast" to the set K

In other words, there is some constant 0 < r < 1 such that after k steps of this algorithm (not the random process), the distance from the k^th approximation to the Sierpinsky triangle is at most r^k

Actual convergence rates of the corresponding random process are more complicated. One of my colleagues recently wrote a paper on convergence rates of the "chaos game" (essentially what this is doing), and one can get really sharp information on how fast the random process converges [3]. However, this uses some non-trivial covering time theory for Markov processes. It's not too complex to follow, but definitely way more detail than I can write up in this note here.

[1] https://en.wikipedia.org/wiki/Hausdorff_distance

[2] https://en.wikipedia.org/wiki/Banach_fixed-point_theorem

[3] https://arxiv.org/abs/2007.11517

Edit: To see that the points converge to the Sierpinsky triangle (with no quantitative information) is a "bit less challenging" (i.e. within the realm of "standard" material - not necessarily easier!) - you can essentially reduce this problem to an application of the Birkhoff ergodic theorem [4]. Firstly, we can consider this process as a random map

pi: {1,2,3}^{\mathbb{N}} -> R^2

which is defined by taking finite subsequences (i1, i2, ..., ik) and mapping them to the images fi1(fi2( ... (fik(x_0))), and taking the limit in k, for some fixed starting point x0.

Then the "apply map fi" can be interpreted as looking at pi(sigma(i1,i2,...)) where sigma is the left-shift map sigma(i1, i2, ...) = (i2, i3, ...). But the shift map plays very nicely with the measure on the sequence space (the Bernoulli measure), in the sense that the Birkhoff ergodic theorem can be applied. This gives that for "typical" sequences of random function applications, this process will converge to the Sierpinsky triangle.

Here's another way to see it: let's say I apply maps fi1, fi2, ..., fik to my starting point x0. Then if we code the "level k" approximations to the Sierpinsky triangle in the same way [triangle (i1, ..., ik) = fi1(...(fik(initial triangle)))], the image of the point x0 will lie in triangle (i1, ..., ik). But these triangles, for large k, approximate the Sierpinsky triangle. Some more argumentation is required to deal with the case when the starting point does not sit in the Sierpinsky triangle, but this isn't a problem because of the fast convergence result explained above (if you wait a long time, all the points will be "very close"). The above ergodic theory argument is just showing that every possible finite subsequence will show up for "typical" random function applications.

[4] https://en.wikipedia.org/wiki/Ergodic_theory

f1(x,y) = (x/2, y/2)

f2(x,y) = (x/2+1/2, y/2)

f3(x,y) = (x/2 + 1/4, y/2 + sqrt(3)/4)

which are precisely the maps in the Iterated Function System [1] which generate the Sierpinsky triangle!

It's a very nice observation that the three maps have a concise representation in terms of taking the midpoint with the given point and the vertices of the triangle.

[1] https://en.wikipedia.org/wiki/Iterated_function_system

I am asking this question in general, though I am motivated by my following (hopefully representative) problem.

I've been trying to obtain a domain authorization code from AWS in order to transfer to a new domain registrar. However, requesting such a code just returns the error

   "Sorry, but an unexpected error has occurred while getting your auth code. Please retry again..."

Of course, since I am not paying for an AWS service plan, I don't expect dedicated customer support. However, the service provided is not meeting the basic requirements. In fact, since I cannot obtain the authorization code, I cannot even transfer the domain to another registrar (which is the other response to an unsatisfactory product!)

[1] https://forums.aws.amazon.com/thread.jspa?threadID=347652

Comments URL: https://news.ycombinator.com/item?id=29459285

Points: 1

# Comments: 0

The research group maintains an excellent webpage with many resources about the TSP [1]. They have also developed an app (Concorde TSP) [2] which provides really good graphical representation of the algorithm. There is an iOS app as well under the same name which has a nice interface.

[1]: http://www.math.uwaterloo.ca/tsp/ [2]: http://www.math.uwaterloo.ca/tsp/concorde/index.html