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Positive dyadic density for rational weighted binary expansions

The paper proves that if a weighted binary expansion $\eta_S = \sum_{n\in S} n 2^{-n}$ is rational, then the index set $S$ must occupy a positive proportion of every sufficiently large dyadic block, a result that resolves Erdős Problem 260 by establishing a local density obstruction to rationality through an analysis of carry states and gap windows.

Original authors: Han Wang

Published 2026-07-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Han Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about numbers. In the world of mathematics, there is a special club called "Rational Numbers." These are numbers that can be written as simple fractions, like 1/2 or 3/4. They are the "neat" numbers of the math world. Then there are the "Irrational Numbers," like $\pi$ or $\sqrt{2}$ . These are the messy, never-ending, non-repeating numbers that go on forever without a pattern.

For a long time, mathematicians have been trying to figure out how to tell if a specific, complicated sum of numbers belongs to the "neat" club or the "messy" one. Imagine you have a machine that adds up a list of numbers. If the machine stops and gives you a clean fraction, it's rational. If it keeps going forever without ever settling into a repeating pattern, it's irrational. The big question is: Can we predict the answer just by looking at where the numbers in the list are placed? If the list is very sparse (meaning the numbers are far apart), does that force the final sum to be messy (irrational)? This is the heart of a famous puzzle known as Erdős Problem 260, named after the legendary mathematician Paul Erdős, who loved asking questions about how numbers behave when they are spread out.

Now, meet a new detective named Han Wang. In a recent paper, Wang tackles a specific version of this puzzle involving "weighted binary expansions." Think of this as a special way of building numbers using powers of two (like 1/2, 1/4, 1/8), but with a twist: each piece has a weight based on its position. Wang's job is to prove that if you try to build a "neat" fraction (a rational number) using these weighted pieces, the pieces cannot be too far apart. In fact, they must be packed together in a very specific, dense way. If the pieces are too sparse, the sum must be messy (irrational). Wang doesn't just guess this; he builds a mathematical fortress to prove it beyond any doubt.

The Mystery of the Sparse Gaps

Let's break down Wang's investigation. Imagine you are building a wall out of bricks. Each brick represents a number in your list. To make a "neat" wall (a rational number), the bricks have to fit together perfectly. Wang discovered a hidden rule: if your wall is made of rational bricks, you can't have huge gaps between them.

He proved that in any large enough section of your wall, the bricks must occupy a "positive proportion" of the space. This means that no matter how far you look down the wall, you will always find a certain minimum amount of bricks. You can't have a section of the wall that is almost entirely empty. If you try to build a wall where the bricks are very far apart (a "sparse" sequence), the math forces the wall to collapse into a messy, irrational shape.

The Detective's Toolkit: Carries and Windows

How did Wang prove this? He used a clever trick involving "carries," which is a concept you might know from adding numbers on paper. When you add 5 and 7, you get 12. You write down the 2 and "carry" the 1 over to the next column. In Wang's math world, these carries are like little messengers that travel down the line of numbers.

Wang showed that if the final sum is a neat fraction, these carry messengers have to follow strict rules. They can't wander off too far; they are bounded by a simple limit. This is the first clue.

The second part of the investigation involves looking at "windows." Imagine sliding a window frame over your wall of bricks. Inside this window, you count how many bricks you see. Wang looked at what happens if you try to make a window that is almost empty (a "sparse" block). He found that if you have a window with very few bricks, the "carry messengers" get confused. They start piling up in a way that creates a huge, impossible imbalance.

To catch this imbalance, Wang used a method called "integrated excess." Think of it like measuring how much water spills out of a bucket if you tilt it. If the bricks are too sparse, the "water" (the mathematical error) spills out so much that it breaks the laws of physics for rational numbers. The math simply cannot balance the equation if the gaps are too big.

The Affine Line: A Magic Train Track

Here is where the story gets really cool. Wang noticed that when these "carry messengers" repeat a certain pattern of gaps, they line up perfectly on a straight line. In math, this is called an "affine line." Imagine a magic train track where the trains (the numbers) must travel. If the gaps between the stops are too long and too regular, the trains are forced onto a single track.

Wang counted how many trains could fit on this track. He found that if the gaps are sparse, the track becomes so crowded with impossible scenarios that it creates a contradiction. It's like trying to fit too many people into a room that is too small; eventually, someone has to break the rules. In this case, the rule is that the sum must be a neat fraction. Since the math breaks, the sum cannot be neat. It must be irrational.

The Verdict

Wang's paper is a complete proof. He didn't just simulate this on a computer or suggest it might be true; he built a logical argument that leaves no room for doubt. He proved that for any infinite list of numbers where the gaps grow large enough (specifically, where the $n$ -th number is much larger than $n$ ), the sum of the series $\sum \frac{a_n}{2^{a_n}}$ is guaranteed to be irrational.

This solves a specific, long-standing version of Erdős Problem 260. It tells us that "neat" numbers have a secret requirement: they need a steady supply of ingredients. You can't make a perfect fraction out of a recipe where the ingredients are scattered too far apart. If the ingredients are too sparse, the result is inevitably messy and infinite.

So, the next time you see a list of numbers stretching out into the distance, remember Wang's finding: if the gaps get too wide, the number you are building will never be a simple fraction. It will be one of those beautiful, endless, irrational numbers that keep mathematicians busy and curious forever.

Technical Summary: Positive Dyadic Density for Rational Weighted Binary Expansions

Problem Statement
The paper addresses the irrationality of weighted binary series of the form $\eta_S = \sum_{n \in S} n 2^{-n}$ , where $S \subseteq \mathbb{N}$ is an infinite set of positive integers. Specifically, it investigates Erdős Problem 260, which asks whether the series $\sum_{n \ge 1} a_n 2^{-a_n}$ must be irrational if the sequence $a_n$ satisfies the growth condition $a_n/n \to \infty$ .

While previous results established irrationality under stronger hypotheses—such as $a_{n+1} - a_n \to \infty$ (Erdős) or $\limsup (a_{n+1}-a_n)/\log a_{n+1} = \infty$ (Borwein and Loring)—the condition $a_n/n \to \infty$ only guarantees global sparsity ( $\# \{n : a_n \le X\} = o(X)$ ). It permits highly irregular local gap patterns, including clusters of short gaps, which previous methods could not rule out. The core difficulty is that a globally sparse sequence might still contain complicated short-gap patterns that evade standard recurrence criteria.

Methodology and Proof Strategy
The paper proves a stronger local result: if $\eta_S$ is rational, then $S$ must occupy a positive proportion of every sufficiently large dyadic block $(X, 2X]$ . This "positive dyadic density" contradicts the sparsity implied by $a_n/n \to \infty$ .

The proof proceeds by contradiction, assuming $\eta_S = P/Q \in \mathbb{Q}$ and that a dyadic block is sparse (i.e., $S$ has low density in $(X, 2X]$ ). The argument combines arithmetic properties of rational tails with combinatorial counting of gap patterns:

Integral Carry Recurrence: Rationality implies that the "tail" of the series can be expressed as a non-negative integer carry $R_N$ satisfying a linear recurrence $R_{N+1} = 2R_N - Q(N+1)d_{N+1}$ with a linear size bound $0 \le R_N \le Q(N+2)$ .
Sparse Block Lower Bound: If a dyadic block were sparse, the gaps between support points would be large. The authors define "window-threshold pairs" $(k, T)$ , where $k$ anchors a window of consecutive gaps and $T$ is a threshold. They define an "integrated excess" $y(k, T)$ representing how much the window span exceeds $T$ . A sparse block forces many windows to have large excess, establishing a lower bound on the total integrated excess.
Combinatorial Upper Bound: The authors analyze the structure of windows with large excess.
- Prefix Selection: They isolate a unique "initial long prefix" of gaps for each large-excess window.
- Lattice Collinearity: Using the carry recurrence, they show that occurrences of frequent prefix patterns must align on affine lines in the $(x, R_x)$ plane (lattice collinearity).
- Slope Dynamics: The evolution of the normalized slope $\mu = K/(QH)$ $μ = K / (Q H)$ along these lines is governed by the map $\mu \mapsto 2^g \mu - 1$ $μ \mapsto 2^{g} μ - 1$ . The authors split the analysis into two regimes:
  - Interior-slope ( $0 < \mu < 1$ ): The slope remains bounded. The authors show that the reduced denominator of the slope eventually stabilizes to a fixed odd integer $q$ . They partition the gap sequence into "logarithmic blocks" and use entropy bounds and Farey separation to show that the number of such occurrences is small.
  - Exterior-slope ( $\mu \notin [0, 1]$ ): Once the slope leaves $[0, 1]$ , its distance from the interval grows exponentially with the span of shared gap words. This exponential separation drastically limits the number of valid integer parameters on the affine line.
Contradiction: The combinatorial analysis yields an upper bound on the total integrated excess that is asymptotically smaller than the lower bound derived from the sparse-block hypothesis. This contradiction proves that a sparse dyadic block cannot exist if $\eta_S$ is rational.

Key Contributions and Results

Theorem 1.1 (Positive Dyadic Density): If $S$ is infinite and $\eta_S = P/Q \in \mathbb{Q}$ (in lowest terms), there exists a constant $c_{dens}(Q) > 0$ such that for every sufficiently large dyadic scale $X = 2^L$ :
$A_S(2X) - A_S(X) \ge c_{dens}(Q) X$
where $A_S(X)$ is the counting function of $S$ . This establishes that rationality forces a uniform positive density in every large dyadic block, not just on a subsequence.
Corollary 1.2 (Resolution of Erdős Problem 260): If $a_1 < a_2 < \dots$ satisfies $a_n/n \to \infty$ , then $\sum_{n \ge 1} a_n 2^{-a_n}$ is irrational.
- Reasoning: The condition $a_n/n \to \infty$ implies $A_S(X) = o(X)$ , which contradicts the positive density lower bound required for rationality.
Methodological Innovation: The paper introduces a technique using overlapping gap windows and integrated excess to handle arbitrary mixtures of short and long gaps without requiring an "exceptional" large gap. It leverages the interplay between the linear bound on carries and the affine geometry of lattice points to count occurrences of gap patterns.

Significance and Claims
The paper claims to resolve Erdős Problem 260 by identifying a local density obstruction to rationality. The result is stronger than previous irrationality criteria because it does not rely on the existence of a subsequence of large gaps; rather, it demonstrates that rationality is incompatible with any local sparsity in dyadic blocks.

The authors explicitly state that their method is tailored to the specific arithmetic structure of weighted binary series (powers of 2 in the denominator and linear weights in the numerator). They do not claim to provide a general solution for other bases, other weight sequences, or a sharper quantitative density law beyond the existence of the constant $c_{dens}(Q)$ . The proof relies on elementary but intricate arithmetic and combinatorial estimates, avoiding deep transcendental number theory in favor of structural constraints on the carry recurrence.

The Mystery of the Sparse Gaps

The Detective's Toolkit: Carries and Windows

The Affine Line: A Magic Train Track

The Verdict

Technical Summary: Positive Dyadic Density for Rational Weighted Binary Expansions

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