A piece from the mathematics of Ted Kaczynski

[mklabgr]

This paper is particularly interesting in relation to Ted Kaczynski, because it is not a paper about his life or political ideas; it is a modern mathematical paper that takes several problems and conjectures from his 1967 mathematics PhD thesis and later mathematical correspondence and proves new results about them.

The paper is Parker Kuklinski, “An uncountable union of line segments with null two-dimensional measure” (2023). Its central problem is geometrical/measure-theoretic: how can one connect two sets of points by an enormous collection of line segments while making the resulting planar set have surprisingly small—or even zero—area?

1. The basic mathematical problem
Imagine two horizontal line segments:
$X_0={(x,0):0\leq x\leq1}$
and
$X_1={(x,1):0\leq x\leq1}$.
Now connect points on $X_0$ to points on $X_1$ with straight line segments.
You might intuitively expect that if you connect every point of the bottom interval to a corresponding point of the top interval, the resulting collection of lines should occupy a substantial amount of the square.
Kaczynski studied precisely this kind of object, which the paper calls an unrestricted trapezoid. The surprising possibility is that an uncountable collection of line segments can have zero two-dimensional Lebesgue measure.
In other words, you can have something that contains an enormous number of curves and nevertheless has
$\mu_2(T)=0$.
That counterintuitive phenomenon is the central theme.

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2. Kaczynski's original contribution
This is where the paper becomes particularly interesting.
Kaczynski's 1967 PhD thesis, “Boundary Functions,” contained a theorem concerning unions of line segments that the new paper builds upon. The authors explicitly cite his thesis as reference [1].
Kaczynski considered a trapezoid
$T=\bigcup \ell$
where each line segment $\ell$ connects a point of $X_0$ with a point of $X_1$.
For the special case where the line segments are disjoint, he proved

$\mu_2(T)=\frac{\mu_1(T_0)+\mu_1(T_1)}{2}$

This is a remarkably clean relationship between the two-dimensional measure of the interior construction and the one-dimensional measures of its boundary sets.
He then used this machinery to construct a collection of non-intersecting paths whose union has measure zero, even though the paths correspond to every point of an interval.
The paper explains that this construction also gives an example of a measurable function whose boundary function is non-measurable.
That is an important distinction:

Kaczynski's mathematical work here was not about technology or politics. It was genuine research in real analysis, measure theory and complex analysis.

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3. The really strange phenomenon: zero boundary measure but huge interior measure
Kaczynski also considered what happens if the line segments are allowed to intersect.
The result becomes dramatically different.
Kuklinski reproduces a result from Kaczynski showing that there is an unrestricted trapezoid with
$\mu_1(T_0)=\mu_1(T_1)=0$
but
$\mu_2(T)=\infty$.
So the two boundary sets can have zero one-dimensional measure, while the union of the connecting segments has infinite two-dimensional measure.
This demonstrates that once intersections are permitted, the intuitive relationship between the boundary and interior breaks down completely.
And this leads directly to the question that Kuklinski investigates:
Can the opposite happen?
Can the boundary sets have large measure while the union of the connecting segments has zero area?

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4. The problem Kaczynski left behind
Kaczynski later returned to this problem, decades after his PhD.
The paper says that more than 30 years later, he reformulated the problem using increasingly thin parallelograms instead of infinitely thin line segments and proposed three conjectures.
Define
$\mu_2^*(T)=\frac{\mu_1^*(T_0)+\mu_1^*(T_1)}{2}$
where $T_n^\sigma$ is constructed from $n$ thin parallelograms according to a permutation $\sigma$.
Kaczynski proposed:
Conjecture 1
$\lim_{n\to\infty}\alpha(n)=0$.
That means that with increasingly fine constructions, the minimum possible area should approach zero.
The fascinating part is that Kaczynski claimed to have proved this. Kuklinski supplies a different proof.
Conjecture 2
Kaczynski suggested that the decay should essentially behave like
$\frac{1}{\log n}$
up to constants.
More specifically, the paper records his proposed form
$c_-\frac{1}{\log n}<\alpha(n)<c_+\frac{1}{\log n}$
and notes that Kaczynski claimed a weaker upper bound than this.
Conjecture 3
Kaczynski conjectured that
$\alpha(n)$
is monotonically decreasing.
This remained open in the formulation discussed by the paper.

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5. What Kuklinski actually proves
This is the main mathematical contribution of the 2023 paper.
The author connects Kaczynski's trapezoid problem to the geometry of the Sierpiński gasket.






The connection is quite elegant.
Kuklinski constructs a function based on ternary expansions. Essentially, digits $1$ and $2$ are exchanged in a base-3 representation:
$1\leftrightarrow2$.
This produces a highly discontinuous/fractal correspondence $f:[0,1]\to[0,1]$.
The associated collection
$S={((1-t)x+tf(x),t):x,t\in[0,1]}$
can nevertheless have
$\mu_2(S)=0$.
That is the key result.
So the paper constructs an uncountable union of line segments connecting two full intervals, yet the entire planar set has zero area.
That's the result behind the paper's title.

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6. Why the Sierpiński gasket appears
This is probably the most beautiful part mathematically.
The ternary construction naturally generates Cantor-set-like slices.
Those slices can be related to projections of the Sierpiński gasket.
The paper then uses a quantity called the Favard length/distance, essentially an average of the lengths of projections of a planar set:
$\operatorname{Fav}(E)=\frac{1}{\pi}\int_0^\pi
\mu(\operatorname{proj}_\theta(E)),d\theta$.
The authors use known estimates concerning projections of finite approximations to the Sierpiński gasket.
The important chain of ideas is:
Kaczynski's trapezoids
$\downarrow$
ternary/fractal construction
$\downarrow$
Cantor-type sets
$\downarrow$
Sierpiński gasket
$\downarrow$
projection lengths
$\downarrow$
Favard length
$\downarrow$
area of the trapezoid tends to zero.
That's the modern mathematical machinery used to attack Kaczynski's old problem.

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7. The main new bound
The paper obtains
$\mu_2(T_{3^n}^{\sigma})\leq Cn^{-p}$
for some constants $C,p>0$.
Then, for general $n$, it obtains the weaker but still significant result
$\alpha(n)<\frac{C}{(\log n)^p}$.
Consequently,
$\boxed{\lim_{n\to\infty}\alpha(n)=0}$.
Thus Kaczynski's first conjecture is established by the paper's argument, although through a different route from Kaczynski's claimed proof.
Importantly, the paper does not prove the full sharp $1/\log n$ conjecture recorded from Kaczynski. Instead it establishes a polynomial-in-$1/\log n$ upper bound.

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8. The connection to Kaczynski's PhD work
There's a deeper historical point here.
Kaczynski's mathematical career had two quite different stages.
1960s:
He was working in complex analysis, geometric measure theory and boundary behavior of functions. His 1967 thesis was concerned with boundary functions and unusual behavior of functions near the boundary of domains.
Later decades:
He returned to some of the geometrical questions from his earlier work and reformulated them in terms of finite collections of parallelograms, producing the conjectures discussed in this paper.
Kuklinski's paper effectively creates a bridge:
Kaczynski 1967 → Kaczynski's later conjectures → modern fractal geometry → new proof.
That is probably the most interesting aspect of the paper if you're specifically studying Kaczynski.

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9. One particularly striking detail
Kaczynski's mathematical work is sometimes overshadowed by his later crimes and manifesto. This paper gives a concrete example of why the distinction between “Kaczynski the mathematician” and “Kaczynski the terrorist” matters.
The paper's bibliography treats his 1967 thesis simply as a mathematical source:

Theodore John Kaczynski, Boundary Functions, PhD thesis, University of Michigan, 1967.

And the author is able to take a problem originating in that thesis and connect it to modern fractal geometry and harmonic/measure-theoretic ideas.
So, mathematically speaking, Kaczynski's relevant legacy here isn't the manifesto. It is a set of unusual questions about measure, boundary behavior and geometric constructions that survived long enough to become useful starting points for later research.

Key takeaways

• Kaczynski's 1967 PhD thesis is the starting point for the mathematical problem.
  
• He studied boundary functions, measure and unusual geometric constructions.
  
• He proved a striking theorem relating the measure of certain trapezoidal unions to their boundary measures.
  
• He later formulated the problem using thin parallelograms.
  
• He proposed three conjectures concerning the minimum possible area $\alpha(n)$.
  
• Kuklinski proves that $\alpha(n)\to0$, establishing the substance of Kaczynski's first conjecture through a different argument.
  
• The proof uses Cantor-type sets, ternary expansions, the Sierpiński gasket and Favard-length estimates.
  
• The resulting construction gives an uncountable union of line segments with zero two-dimensional measure.
  
• The paper therefore represents a genuine example of a modern mathematical development growing out of one of Kaczynski's old research problems.
  

One correction to a common impression is important: this paper does not show that Kaczynski was secretly a major modern mathematician whose later work was ignored. It shows something more precise and interesting: he posed some genuinely nontrivial mathematical questions, and at least one of those questions has subsequently been connected to modern fractal geometry and solved in a new way.

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