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Small sums of roots of unity

This paper establishes that non-vanishing sums of $k$ $N$ -th roots of unity can be bounded above by $N^{-\alpha_k}$ , where the exponent $\alpha_k$ grows with $k$ , and further investigates improved bounds for a positive proportion of $N$ .

Original authors: Alexandros Kalogirou

Published 2026-07-08

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexandros Kalogirou

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 have a giant, perfect clock face with N tiny marks on the edge, representing the "roots of unity." If you pick a few of these marks and draw a line from the center to each one, you have a set of arrows.

Now, imagine you add all those arrows together. Usually, they cancel each other out, and the final result is zero (like a tug-of-war where everyone pulls equally hard in different directions).

The Big Question:
What is the smallest non-zero result you can get if you are forced to use exactly k arrows? How close to zero can you get without actually hitting zero?

This paper, written by Alexandros Kalogirou, is a mathematical detective story trying to find the answer to that question. The author wants to know: If you pick a specific number of arrows (k), how tiny can the leftover "wobble" be?

Here is the breakdown of the paper's findings using simple analogies:

1. The Goal: Finding the "Tiniest Wobble"

The author defines a function, let's call it f(k, N).

k is the number of arrows you must use.
N is the total number of marks on the clock (how "fine" the clock is).
f(k, N) is the size of the smallest non-zero leftover sum.

The paper asks: As the clock gets infinitely detailed (N gets huge), how small can that leftover wobble get?

2. The Old Rules vs. New Discoveries

Before this paper, mathematicians knew some rules. For example, if you have 5 arrows, the leftover wobble gets smaller as the clock gets bigger, but only at a certain speed.

The Author's New Discovery:
The author shows that for larger numbers of arrows (k), the leftover wobble can get much, much smaller than previously thought.

Think of it like this: If you have 4 or more arrows, you can arrange them so they almost perfectly cancel out, leaving a wobble that is incredibly tiny—so tiny that it shrinks faster than the clock gets bigger.
The more arrows you have (the higher k is), the better you can make them cancel out. The "wobble" shrinks at a rate that gets faster and faster as you add more arrows.

3. The Special Case of 7 Arrows (The "Magic Trick")

The paper highlights a specific case: k = 7.

The Claim: With 7 arrows, you can make the leftover wobble so small that it is roughly the size of $1 / N^{1.5}$ .
The Metaphor: Imagine trying to balance 7 people on a seesaw. The author found a specific arrangement where, even if the people are standing on a very fine grid, the seesaw tilts so slightly that it's almost perfectly flat. This is a significant improvement over what was known before.

4. How Did They Do It? (The "Perturbation" Method)

The author didn't just guess; they used a clever strategy called perturbation.

The Idea: First, find a way to arrange the arrows so they cancel out perfectly (result = 0). But wait, the problem says the sum must not be zero.
The Trick: Take that perfect zero arrangement and nudge the arrows just a tiny, tiny bit.
- If you nudge them randomly, the result might be huge.
- But the author used advanced math (like a "fine-tuning" tool) to nudge them in a very specific way. They nudged the arrows just enough to break the perfect zero, but not enough to create a big wobble.
The Result: This "nudge" creates a sum that is non-zero, but incredibly close to zero.

5. The "Dense" Clocks (Special Numbers)

The paper also notes that if the number of marks on the clock (N) has certain special properties (like being a multiple of 6 or 10), you can do even better.

Analogy: It's like saying, "If your clock has a number of marks that is divisible by 6, you can arrange 5 arrows to cancel out even more perfectly than if the clock had a random number of marks."
For example, if N is a multiple of 6, the leftover wobble for 5 arrows becomes even tinier than the general rule suggests.

Summary of the Main Takeaways

Better Bounds: The author proved that for a group of arrows (k), the smallest possible non-zero sum shrinks much faster than we used to think as the clock gets bigger.
The More, The Merrier: As you add more arrows to your group, your ability to make them cancel out improves dramatically.
Specific Wins: The paper gives specific "best-case" scenarios for groups of 7, 11, and other numbers, showing exactly how tiny the wobble can get.
The Method: The secret sauce is taking a perfect zero-sum arrangement and "nudging" it just enough to make it non-zero, using math to ensure that "nudge" doesn't create a big mess.

In short, this paper tells us that nature (or at least, the math of these numbers) allows for incredibly precise balancing acts. If you have enough arrows, you can make them cancel out to a degree that is almost unimaginably small.

Technical Summary: Small Sums of Roots of Unity

Problem Statement
The paper addresses the problem of determining the magnitude of the smallest non-vanishing sum of $N$ -th roots of unity consisting of $k$ terms. Formally, let $f(k, N)$ denote the modulus of the smallest non-zero sum:
$f(k, N) = \min \left\{ |S| : 0 \neq S = \sum_{i=1}^k \zeta_i, \text{ with } \zeta_i^N = 1 \right\}.$
The primary objective is to establish upper bounds for $f(k, N)$ as $N \to \infty$ . While previous work by Myerson established bounds of the shape $O(N^{-k/4})$ for even $k$ and $N$ , and Barber improved this to $O(N^{-4/3})$ for $k=5$ , the general question of whether bounds of the form $O(N^{-\alpha})$ with $\alpha > 1$ exist for all $k$ and $N$ remains largely open.

Methodology
The paper employs a perturbation strategy, building upon the approach introduced by Barber. The core idea involves starting with a configuration of roots of unity that sums to zero (a vanishing sum) and perturbing the arguments slightly to generate a non-vanishing sum with a small modulus.

Key technical components include:

Inhomogeneous Diophantine Approximation: The proofs rely heavily on Lemma 1 (derived from Bugeaud and Laurent), which provides uniform bounds for linear forms. This allows the authors to find integer vectors $\vec{x}$ such that linear combinations of roots of unity approximate zero within a specific error margin.
Calculus and Implicit Function Theorem: To handle specific cases (notably $k=7$ ), the authors utilize the Implicit Function Theorem to demonstrate that certain maps involving sine and cosine functions are local diffeomorphisms. This ensures that the image of the map contains an open neighborhood, allowing the construction of specific algebraic relations among the roots.
Combinatorial Construction (Frobenius Coin Problem): For the general case, the authors construct sums by combining powers of $(1+\zeta)$ and $(1+\omega+\omega^2)$ , where $\zeta$ and $\omega$ are roots of unity close to $-1$ and $e(1/3)$ respectively. This leverages the Frobenius coin problem to represent $k$ as a linear combination of 2 and 3.
Case Analysis: The proof for $k=7$ distinguishes between cases where the vanishing sum has integer solutions (rational angles) and cases where it does not, utilizing Taylor expansions to estimate the error terms introduced by the perturbation.

Key Contributions and Results

Theorem 1 ( $k=7$ ): The paper proves that $f(7, N) = O(N^{-3/2})$ . This is achieved by perturbing a specific vanishing sum configuration derived from the equation $1 + \sum_{i=1}^3 [e(\alpha_i) + e(-\alpha_i)] = 0$ . The proof involves showing that the perturbation can be controlled to yield a sum of order $N^{-3/2}$ while ensuring the sum remains non-zero.
Theorem 2 (General $k$ ): The authors establish a general upper bound:
$f(k, N) = O(N^{-\lfloor \log_6 k \rfloor}).$
This result implies that for $k \ge 4$ , the exponent $\alpha_k$ in the bound $O(N^{-\alpha_k})$ satisfies $\alpha_k > 1$ , and crucially, $\alpha_k \to \infty$ as $k \to \infty$ .
- The paper notes that the same method yields $f(k, N) = O(N^{-2})$ for $11 \le k \le 36$ (with $k \neq 11$ ).
- For $k=11$ , a similar proof to Theorem 1 yields $f(11, N) = O(N^{-5/3})$ .
Improvements for Specific $N$ : The final section demonstrates that for $N$ lying in dense sets (specifically multiples of certain integers), the exponents can be improved:
- If $6 \mid N$ , then $f(5, N) \ll N^{-2}$ .
- If $10 \mid N$ , then $f(7, N) \ll N^{-2}$ .
- If $2 \mid N$ , then $f(10, N) \ll N^{-4}$ . This result for $k=10$ matches the best known result by Myerson for even $N$ but is derived via a construction involving linear equations rather than the Prouhet-Tarry-Escott problem.

Significance and Claims
The paper claims to provide a double goal achievement:

It demonstrates that for general $N$ , the decay rate of the smallest non-vanishing sum improves significantly as $k$ increases, with the exponent $\alpha_k$ growing without bound.
It shows that for a positive proportion of $N$ (multiples of specific numbers), even sharper bounds can be achieved.

The work is presented as a continuation and generalization of the methods introduced by Barber, specifically adapting the perturbation technique to handle higher $k$ and specific algebraic structures. The authors note that the proofs rely on perturbing configurations of roots of unity to ensure the resulting sum is non-vanishing, a step justified by theorems regarding the separation of angles in vanishing sums (Conway and Jones). The paper does not claim to solve the problem for all $k$ and $N$ with a single universal exponent but rather provides a family of bounds that improve with $k$ .