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Note on unique representation bases

This paper improves the lower bound of the constant $c_{\mathscr{A}}$ for the density of a unique representation basis of $\mathbb{Z}$ from $\sqrt{2}/2$ to $1$.

Original authors: Yuchen Ding, Jie Wang

Published 2026-02-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuchen Ding, Jie 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

The Mystery of the Perfect Puzzle: A Simple Guide to "Unique Representation Bases"

Imagine you are a master puzzle maker. You have a set of numbered tiles (like 1, 5, 10, etc.), and your goal is to create a "Perfect Sum Set."

In a Perfect Sum Set, every single integer—positive or negative—must be formed by adding exactly two of your tiles together. But there is a catch: there can be only one way to make each number. If you can make the number "7" by adding $3+4$ , you are forbidden from making "7" any other way (like $1+6$ or $2+5$ ).

In mathematics, this "Perfect Sum Set" is called a Unique Representation Basis.

The Big Question: How "Crowded" Can the Tiles Be?

Mathematicians have been wondering: if you want to make sure you can form every number using only one specific combination, how many tiles do you need to keep in your toolbox?

If you have very few tiles, you won't be able to make all the numbers. If you have too many tiles, you’ll accidentally create the same number in multiple ways (like making "10" with $5+5$ AND $7+3$ ), which breaks the "uniqueness" rule.

The researchers in this paper are looking for the "sweet spot." They want to know the maximum density of these tiles. Specifically, if you look at all your tiles within a certain range (say, between $-x$ and $x$ ), how many can you fit before the system breaks?

The "Goldilocks" Problem

For a long time, mathematicians knew that these sets couldn't be too crowded. There is a mathematical limit (a "speed limit") for how many tiles you can have without causing "collisions" (duplicate sums).

Before this paper, we knew the answer was somewhere between a certain value (roughly $0.707$) and a maximum limit (roughly $1.414$). It was like knowing a person's height is somewhere between 5 feet and 7 feet, but not knowing if they are a basketball player or a toddler.

This paper proves that the "sweet spot" is at least 1. They have narrowed the range significantly, moving the floor up from $0.707$ to $1$.

How They Did It: The "Building Block" Strategy

The authors used a clever "Lego-style" construction method. Instead of trying to find the perfect set all at once, they built it in stages:

The Foundation (Small Sets): They start with a tiny, simple set of tiles that works for small numbers.
The Expansion (The Sidon Set): To grow the set without breaking the rules, they use something called a Sidon Set. Think of a Sidon Set as a "Socially Distanced Set." In this set, every pair of numbers is so uniquely spaced out that their sums never overlap. It’s like a group of people standing so far apart that no two pairs can ever form the same distance between them.
The Inductive Leap: They use a mathematical technique called induction. They prove that if they have a working set for small numbers, they can always find a "socially distanced" group of new tiles to add that will cover the next batch of numbers without ever creating a duplicate sum.

The Takeaway

The authors have shown that you can actually build a very "dense" collection of tiles—one that is quite large and efficient—while still maintaining the strict rule that every number has one, and only one, unique "recipe."

They conclude by suggesting that the absolute best possible density is actually $\sqrt{2}$ (about $1.414$), but for now, they have successfully pushed the boundary higher than anyone else, proving that these "Perfect Sum Sets" can be much more robust than we previously thought.

Technical Summary: Note on Unique Representation Bases

Authors: Yuchen Ding and Jie Wang
Subject Area: Additive Number Theory (Sidon sets, representation bases, growth of sets)

1. The Problem

The paper addresses a specific problem in additive number theory concerning the density and growth of unique representation bases for the integers ( $\mathbb{Z}$ ).

A subset $A \subset \mathbb{Z}$ is a unique representation basis if every integer $n \in \mathbb{Z}$ can be uniquely represented as the sum of two elements $a, a' \in A$ (where $a \leq a'$ ). This is denoted as $r_A(n) = 1$ for all $n \in \mathbb{Z}$ .

The central focus is the growth rate of the counting function $A(-x, x) = |A \cap [-x, x]|$ . Specifically, the authors investigate the constant $c_A$ , defined as the supremum of the values $c$ such that $A(-x, x) \geq c\sqrt{x}$ for infinitely many $x$ . Previous research established that:
$\frac{\sqrt{2}}{2} \leq c_A \leq \sqrt{2}$
The goal of this note is to improve the lower bound of this constant.

2. Methodology

The authors employ an inductive construction process combined with results from the theory of Sidon sets. A Sidon set is a set where all pairwise sums are distinct.

Key components of the methodology include:

Inductive Construction: The authors construct a nested sequence of sets $A_1 \subset A_2 \subset \dots \subset A_h \subset \dots$ $A_{1} \subset A_{2} \subset \dots \subset A_{h} \subset \dots$ and a sequence of integers $x_1 < x_2 < \dots < x_h < \dots$ $x_{1} < x_{2} < \dots < x_{h} < \dots$ . Each step in the induction is designed to satisfy four specific properties:
1. $r_{A_h}(n) \leq 1$ (maintaining the unique representation property).
2. $r_{A_{2h}}(n) = 1$ for all $|n| \leq h$ (ensuring the set eventually covers all integers).
3. $A_{2h-1}(-x_h, x_h) \geq (1-\epsilon)\sqrt{x_h}$ (ensuring high density/growth).
4. $0 \notin A_h$ (avoiding triviality).
Sidon Set Integration: To satisfy the growth requirement (Property III), the authors use a specialized Sidon set $S$ derived from a prime $p$ . They utilize a classical result by Bose to find a large Sidon set within specific intervals of $[-p^2, p^2]$ .
Refinement via Deletion: To ensure the new elements added from the Sidon set do not violate the unique representation property when combined with the existing set $A_{2h}$ , the authors perform a "deletion" step. They remove elements from $S$ that would create duplicate sums when added to the existing elements of $A_{2h}$ .

3. Key Contributions and Results

Improvement of the Lower Bound: The primary result is Theorem 1, which proves that:
$c_A \geq 1$
This improves the previous lower bound of $\sqrt{2}/2 \approx 0.707$ to $1$.
Proof of Existence: The paper provides a constructive proof that there exists a unique representation basis $A$ such that for infinitely many $x$ , the density $A(-x, x)$ is at least $(1-\epsilon)\sqrt{x}$ .
Technical Lemma: The authors prove a new Lemma regarding the existence of a Sidon set $S$ within specific partitioned intervals of $[-p^2, p^2]$ with a cardinality of at least $(1 - \frac{3\epsilon}{4})p$ .

4. Significance

This work advances the understanding of the structural limits of additive bases. While the Erdős-Turán conjecture suggests that additive bases for natural numbers cannot have bounded representation functions if they are "too thin," the behavior in the integers ( $\mathbb{Z}$ ) is fundamentally different.

By pushing the lower bound of $c_A$ toward $1$, the authors bring the mathematical community closer to determining the exact value of $c_A$ . The authors conclude with a conjecture that the true value is $c_A = \sqrt{2}$ , which would imply that unique representation bases can be as dense as the densest possible Sidon sets.