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Counting Algebraic Integers of Bounded Height in Cyclotomic Fields

This paper establishes an asymptotic bound, as the prime power $q$ grows, on the number of algebraic integers and units with absolute height at most $B$ within the cyclotomic fields $\mathbb{Q}[\zeta_q]$ .

Original authors: Phillip Harris, John Yin

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Phillip Harris, John Yin

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 standing in a vast, infinite library. This library isn't filled with books, but with numbers. Specifically, it contains a special type of number called "algebraic integers." These aren't your everyday numbers like 1, 2, or 3; they are complex, multi-dimensional numbers that live in specific mathematical worlds called Cyclotomic Fields.

Think of these fields as different "neighborhoods" in the library. The authors of this paper are focusing on a specific neighborhood called $\mathbb{Q}(\zeta_q)$ , which is built using a special number called a root of unity (imagine a clock face where the hands can point to any fraction of a circle, not just the hours).

The Big Question: How Many Numbers Fit in a Box?

The authors are asking a very specific question: If we put a size limit on these numbers, how many of them can we find in this neighborhood?

To measure the "size" of these numbers, they use a ruler called Height.

The Ruler (Height): Imagine every number has a "loudness" or "complexity." The Height is a measure of how loud or complex a number is.
The Box (Bound $B$ ): The authors say, "Let's only count the numbers that are quieter than a certain volume, $B$ ."

The twist in this story is that usually, mathematicians fix the neighborhood (the field) and ask how the number of items grows as the box gets bigger. Here, the authors do the opposite: They keep the box size ( $B$ ) fixed and ask what happens as the neighborhood gets bigger and more complex.

The Two Types of Numbers They Counted

The authors looked at two different groups of numbers in this neighborhood:

1. The "Units" (The Special Residents)

What are they? These are numbers that have a "partner" that multiplies with them to make 1. They are the VIPs of the number world.
The Cyclotomic Units: Among these VIPs, there is a special subgroup called "Cyclotomic Units." These are the ones that can be built easily from the basic ingredients of the neighborhood (like building a house out of standard bricks).
The Finding: The authors found a formula to estimate how many of these VIPs exist below the volume limit.
- They discovered that as the neighborhood gets huge (as the prime power $q$ grows), the number of these VIPs grows, but not too fast. It grows in a way that is "sub-exponential."
- The Analogy: Imagine a party where the number of guests is limited by how much food ( $B$ ) is available. As the room gets bigger, you can fit more people, but the authors proved that even in a massive room, the number of people doesn't explode into infinity instantly; it grows at a manageable, predictable rate.

2. The "Integers" (The General Population)

What are they? These are all the algebraic integers in the neighborhood, not just the VIPs.
The Finding: The authors also counted the general population. They found that the number of these integers grows exponentially with the size of the neighborhood, but the "loudness" (height) of the numbers keeps them in check.
- The Analogy: If the VIPs are the people in the front row, the integers are everyone in the stadium. The authors calculated that even with a strict volume limit, the total number of people in the stadium is bounded by a specific formula involving the size of the stadium and the volume limit.

How Did They Do It? (The Tools)

To solve this, the authors used some heavy mathematical machinery, which they translated into a few clever tricks:

The Lattice (The Grid): They imagined the numbers as points on a giant, multi-dimensional grid.
The Shape (The Polytope): The condition "height less than $B$ " creates a specific shape in this grid (like a weird, multi-sided ball).
Counting the Dots: The problem became: "How many grid points fit inside this weird shape?"
The Davenport Lemma: This is a famous rule in math that helps estimate how many dots fit inside a shape without having to count them one by one. The authors used this rule to get their upper limits.

The "Class Number" Mystery

There is one small variable in their formula for the VIPs (the units) called $h^+_q$ . This is a number that describes how "messy" the neighborhood is.

The authors admit they can't perfectly control this number.
However, they mention a popular guess (the Cohen-Lenstra heuristics) that suggests this "messiness" number is usually very small or stays constant.
The Takeaway: If this guess is true, then their formula for the VIPs is even more reliable.

Summary in Plain English

The paper is a census report for a specific type of mathematical neighborhood.

The Goal: Count how many special numbers (units) and regular numbers (integers) exist in a neighborhood if we only count the "quiet" ones (those with low height).
The Method: They treated the numbers as points on a grid and used geometric rules to estimate how many points fit inside a specific volume limit.
The Result: They provided a mathematical "ceiling" (an upper bound) for how many of these numbers can exist. They showed that even as the mathematical neighborhood becomes infinitely complex, the number of "quiet" numbers grows in a way that can be predicted and bounded, rather than spiraling out of control.

They didn't find a new medicine or a new technology; they simply drew a more precise map of how these abstract numbers are distributed in the vast landscape of mathematics.

Technical Summary: Counting Algebraic Integers of Bounded Height in Cyclotomic Fields

1. Problem Statement

This work addresses the problem of counting algebraic integers and units within cyclotomic fields $\mathbb{Q}(\zeta_q)$ , where $q = p^r$ is a prime power, subject to a fixed absolute height bound $B$ . Unlike the classical Batyrev-Manin conjecture, which fixes the base field and varies the height bound $B \to \infty$ , this paper fixes the height bound $B$ and investigates the asymptotic behavior of the count as the field degree grows (specifically as $r \to \infty$ ).

The authors define two primary counting functions:

Units: $U(n, B)$ , the number of units in $\mathbb{Z}[\zeta_n]$ with logarithmic Weil height $h(\alpha) < B$ . A subset of interest is $U_{cyc}(n, B)$ , counting only cyclotomic units.
Integers: $I(q, B)$ , the number of algebraic integers in $\mathbb{Z}[\zeta_q]$ with $h(\alpha) \le B$ .

The goal is to establish upper and lower bounds for these quantities as $q$ grows, specifically analyzing the dependence on the field degree $D = \phi(q)$ .

2. Methodology

The proofs rely on a combination of algebraic number theory and geometric lattice point counting.

2.1 Geometric Framework

The core geometric tool is Davenport's Lemma [2], which provides an error bound for the number of lattice points in a compact body $P \subset \mathbb{R}^D$ relative to its volume. The bound depends on the volumes of projections of $P$ onto coordinate subspaces.

For cyclotomic units, the problem is transformed into counting integer vectors $c$ such that $\|Ac\|_{L_1} < T$ , where $A$ is a matrix derived from the logarithms of the absolute values of Galois conjugates of the cyclotomic unit generators.
For algebraic integers, the authors utilize the Minkowski embedding $\Phi: K \to \mathbb{R}^D$ . The set of integers with bounded height is contained within a region $R_1$ defined by logarithmic constraints on the coordinates. This region is approximated by a union of hyper-rectangles (boxes) $Q(k)$ to facilitate volume estimation.

2.2 Algebraic Tools

Cyclotomic Units: The authors utilize the explicit basis of cyclotomic units $\xi_a$ (Proposition 1) and their relation to the full unit group via the class number $h_q^+$ . The action of the Galois group on these units is analyzed using Dirichlet characters.
Spectral Analysis: The matrix $A$ governing the height of units is diagonalized using nontrivial even Dirichlet characters. The eigenvalues $\lambda_\chi$ are expressed in terms of Gauss sums $\tau(\chi_f)$ and $L$ -functions $L(1, \chi_f)$ . Siegel's lower bound for $L$ -functions is employed to estimate the magnitude of these eigenvalues.
Determinant Estimates: For the integer counting problem, the volume of the transformed regions involves the determinant of the Minkowski embedding matrix. The authors use the Cauchy-Binet formula and Hadamard's inequality to bound the sum of squared minors of the matrix, leveraging the known discriminant of cyclotomic fields.

3. Key Results

3.1 Bounds for Cyclotomic Units

Theorem 1 establishes the asymptotic behavior for cyclotomic units in $\mathbb{Q}(\zeta_q)$ where $q=p^r$ is an odd prime power.

Upper Bound: The number of cyclotomic units satisfies:
$U_{cyc}(q, B) \ll_{q^3} \exp\left(C_\epsilon B^{1/3} q^{5/6+\epsilon}\right)$
For the full unit group, the bound includes the class number of the maximal real subfield $h_q^+$ :
$U(q, B) \ll_{q^3} \exp\left(C_\epsilon (h_q^+ B)^{1/3} q^{5/6+\epsilon}\right)$
Lower Bound: Proposition 4 demonstrates that for sufficiently large $B$ , $\log U_{cyc}(q, B) \gg B \log q$ , indicating a polynomial lower bound in $q$ .

The authors note that while $h_q^+$ is difficult to control generally, numerical evidence and Cohen-Lenstra heuristics suggest it is small (potentially constant) for fixed $p$ as $r \to \infty$ , which would imply a subexponential bound for the full unit group in that regime.

3.2 Bounds for Algebraic Integers

Theorem 2 provides an exponential upper bound for the number of algebraic integers of bounded height:
$\log I(q, B) \ll D(B + 1)$
where $D = \phi(q)$ .

Lower Bound: The paper constructs a polynomial lower bound by choosing coefficients in the power basis $\zeta_q^i$ such that the sum of absolute values is bounded, yielding roughly $q^{e^B}$ choices.
Comparison: The authors observe that in the function field analogue $\mathbb{F}_q(t)$ , the count is exactly $q^{B+1}$ , suggesting the true count in the number field case might be polynomial in $q$ , though the proven upper bound is exponential in the degree $D$ .

4. Significance and Claims

The paper claims to provide the first asymptotic bounds for the number of algebraic integers and units of fixed height in cyclotomic fields as the degree of the field tends to infinity.

Contrast with Literature: The work contrasts with previous results (e.g., Dubickas [4]) which fixed the degree and let the height grow. Here, the height is fixed, and the field varies.
Subexponential Conjecture: While the proven bounds for units are exponential in $q^{5/6}$ , the authors conjecture that the true growth is likely polynomial in $q$ (subexponential in the degree) based on the behavior of the class number $h_q^+$ and the function field analogy.
Methodological Contribution: The paper demonstrates the utility of combining Davenport's lattice point counting lemma with explicit spectral properties of cyclotomic units (via Dirichlet characters) and precise determinant estimates for Minkowski embeddings.

The authors maintain a modest tone regarding the optimality of the bounds, acknowledging that the exponential factor in the unit bound arises from the current inability to tightly control the class number $h_q^+$ and the geometric covering arguments used for the integer count.