Unsolved oddities in one dimension

The Besicovitch 1/2 Conjecture is a fascinating topic that has captured the interest of numerous mathematicians, yet remains largely unknown—it's even absent from Wikipedia. At its core, this conjecture aims to quantify the peculiar nature of one-dimensional sets. To clarify these concepts, we’ll begin with some fundamental ideas.

Line Segments

When we think of a “1-dimensional” object, a line segment is the quintessential example. Understanding this concept is crucial for grasping the subsequent sections of this article.

Consider a line segment and a specific point along it. We will examine how much of the line is contained within a circle centered at that point:

As long as the circle is sufficiently small to remain on the line, the length of the line within the circle is 2r. To evaluate “weirdness,” we focus on what occurs at extremely small scales, akin to the tangent line approach in calculus. This involves letting r approach zero, which can be visualized as “zooming in” on the point.

Tracking the radius is essential, as it indicates our current “scale.” This is reminiscent of the limit we define in calculus when determining derivatives. We take the limit of the ratio of the line segment inside the circle divided by 2r. As r approaches 0, we denote this ratio as the density at point p, represented by d(p). In academic literature, you may encounter this term as spherical density since, in higher dimensions, we will use spheres instead of circles.

This computation is straightforward: as long as p is not an endpoint and r is small, the quantity within the circle remains 2r. Therefore, the ratio consistently equals 1, even without taking the limit. Conversely, at an endpoint, the ratio is 1/2.

In summary, we find that the density along the line segment is uniformly 1, except at the endpoints where it measures 1/2.

Nice Curves

Next, we can slightly elevate the complexity by considering differentiable curves, which we often study in calculus. While there are multiple interpretations of this, we will keep it simple and focus on developing our intuition.

If you're ready for higher dimensions, you can visualize it as the image of [0,1] in a space where a tangent line can be defined, or as the graph of a differentiable function in R².

The key point here is that the curve remains 1-dimensional, and, except at the endpoints, it has a single tangent line. The fundamental concept of differential calculus suggests that as we zoom in on a point, the curve increasingly resembles its tangent line.

If you're familiar with calculus, consider proving that the density at any non-endpoint of a differentiable curve is precisely 1, similar to the line segment. At the endpoints, the density will again be 1/2.

It’s crucial to recognize that being differentiable is a stringent condition. However, this developing picture indicates we are on the right path toward defining a meaningful measure of “weirdness.” For smooth curves, the density is consistently 1 almost everywhere, with a few exceptions at isolated points.

Large Density and Crossings

However, don’t be misled—this topic isn’t as straightforward as it initially appears. We aim to create a conjecture that encompasses all “one-dimensional sets,” regardless of their characteristics.

Consider this simple example of two crossing line segments. Clearly, this should qualify as a one-dimensional set. Ignoring the crossing point, we can find sufficiently small circles that avoid it, yielding a density of 1, except at the endpoints, where it remains 1/2.

Nonetheless, for the crossing point, any circle we create will always contain a segment of 4r, leading to a density of 2 there.

This clarifies our understanding. We can construct one-dimensional sets where points exhibit arbitrarily high densities by creating intersections like this.

Reflect on these questions to deepen your understanding: - Must the density always be 1 almost everywhere, with isolated points being 1/2 or more? - Can you conceive of a scenario where the density lies between 1/2 and 1? - Is there a construction where the density exceeds 1 or equals 1/2 for “most” of the set? - Is it possible for the density to be less than 1/2?

Contemplating these scenarios will aid your grasp of the conjecture, but don’t overthink it, as some of these situations are ruled out by established theorems.

This sets the stage for the following section, where we delve into more technical details.

One-dimensional Sets

We define a set as one-dimensional if it possesses a Hausdorff dimension of 1. Although a detailed definition is beyond this article's scope, the concept is often associated with fractals. Fractals can accumulate with intricate edges, causing the Hausdorff dimension to lie strictly between 1 and 2.

For instance, the well-known Sierpinski triangle fractal has a dimension of approximately 1.585, demonstrating how fractals can appear one-dimensional while actually possessing additional dimensions.

The significance of specifying the set with a Hausdorff dimension of 1 lies in its genuine 1-dimensional characteristics in an intuitive sense. These sets can be quite unusual but are not so fractal-like that they acquire extra dimensions.

An example of a set with dimension 1 that cannot be constructed through our prior discussions is the graph of the characteristic function of the rational numbers.

This function plots (x,1) for rational x and (x,0) for irrational x.

While we often visualize this as two lines, in reality, the lower part has gaps throughout, and the upper part is nearly nonexistent. Since the lower segment is still fundamentally a line, it remains fundamentally within our existing understanding.

In terms of density, this means the upper segment has a density of 0, while the lower segment maintains a density of 1.

This example may not elucidate the potential oddities of one-dimensional sets, but it does enhance our understanding of their nature.

It turns out that any one-dimensional set S can be categorized into three distinct components: E, R, and U (these are not standard notations).

• E represents empty sets, which are not genuinely 1-dimensional.
• R stands for rectifiable sets, which are well-behaved, consisting of almost entirely smooth curves. This includes the peculiar characteristic function mentioned earlier.
• U denotes unrectifiable sets, essentially the remainder.

Purely Unrectifiable Sets

Before we proceed, it's essential to grasp the concept of purely unrectifiable one-dimensional sets. Here are two key ideas:

• Regardless of how much you zoom in, you will encounter multiple tangent directions.
• They resemble fractals, but not to the extent that their dimensions exceed 1.

If this suffices for your understanding, feel free to move on. Here’s a comprehensible construction of such a set, known as the four-corner Cantor set. If you're familiar with the Cantor set’s construction, this will feel familiar.

Begin with a square. Divide it into a 4x4 grid, retaining only the four corner squares. Repeat this process with those squares.

Continue this iterative process, and in the limit, you will create a set. While it's not immediately obvious that this results in something one-dimensional or purely unrectifiable, our intuition suggests otherwise. The process generates a fractal-like structure with “corners” everywhere, complicating the definition of a tangent.

The Besicovitch 1/2 Conjecture

Now, let’s address the core of the conjecture you’ve been anticipating. Consider our one-dimensional set S in R², divided into the three components E, R, and U. We can disregard the empty component E as irrelevant.

For our analysis, we accept the rectifiable part R as understood. For instance, Besicovitch demonstrated in 1928 that for almost all x in R, d(x) = 1. This reinforces our earlier intuition.

The pressing question is: how peculiar can the purely unrectifiable component U become?

Recalling our calculus lessons, limits may not always exist. Besicovitch proved in 1938 that for almost all x in U, the limit defining the spherical density fails to exist.

By using a liminf instead of a limit, we arrive at what’s termed the lower spherical density, denoted as d_(x). The minus sign signifies "lower."

Now, a new question arises: what is the smallest number, A, such that d_(x) ? A for almost every x in U (for any purely unrectifiable U)?

You might wonder: Why would anyone inquire about this?

The answer is that it holds practical significance. It serves as a numerical characterization for unrectifiable sets. Since d_(x) = 1 on rectifiable sets (almost everywhere), any A < 1 provides a means to detect unrectifiability.

In his 1938 publication, Besicovitch established that A ? 3/4 and illustrated that A ? 1/2.

The Besicovitch 1/2 Conjecture posits that A = 1/2.