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Power and rank-weighted sums in dense finite Sidon sets

This paper establishes an asymptotic formula for power sums of dense finite Sidon sets within specific residue classes, removing a previous hypothesis by leveraging the Fourier uniformity of extremal Sidon sets.

Original authors: Yuchen Ding

Published 2026-06-16

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuchen Ding

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 jar filled with numbered marbles, ranging from 1 to a huge number $n$ . Now, imagine you are trying to pick out a special group of marbles called a Sidon set.

The rule for this special group is strict: if you pick any two marbles from your group and add their numbers together, that specific sum must be unique. No two different pairs of marbles can add up to the same number. It's like a puzzle where every combination creates a fingerprint that no other combination can copy.

Mathematicians have long been interested in the "dense" versions of these sets—groups that are as full as possible, containing roughly the square root of the total number of marbles available ( $\sqrt{n}$ ).

The Big Question: Are They Fairly Distributed?

The author of this paper, Yuchen Ding, asks a simple but deep question: If you look at these special marbles, are they spread out evenly across the number line, or do they clump together in certain patterns?

Specifically, the paper looks at two ways of checking this:

Residue Classes: If you sort the marbles by what number they leave as a remainder when divided by a number $m$ (like sorting by "even" vs. "odd," or by remainders 0, 1, 2 when divided by 3), are the marbles distributed fairly among these groups?
Weighted Sums: If you don't just count the marbles, but add up their values (or even their values raised to a power, like squaring them), does the total sum match what you would expect if the marbles were perfectly random?

The Previous Problem: A "Conditional" Answer

In earlier work, the author and others found that these sums did look like they were distributed evenly, but there was a catch. The proof relied on a "conditional" assumption: it assumed that the marbles were distributed evenly all the way up to the very end of the list.

However, other mathematicians pointed out that this assumption is like assuming a crowd of people is perfectly spaced out just because they look spaced out in the middle of the room. In reality, the crowd might bunch up right at the exit. Because of this, the previous answer wasn't a guaranteed fact; it was a "if this holds, then that follows" scenario.

The New Discovery: A "Unconditional" Proof

This paper removes that "if." It proves that without needing to assume anything about the very end of the list, the marbles in a dense Sidon set are indeed distributed exactly as expected.

The Analogy of the "Fourier Uniformity":
To prove this, the author uses a tool called "Fourier uniformity," developed by Ortega and Prendiville. Think of this tool as a high-tech scanner that can detect if a pattern is "noisy" or "smooth."

If the marbles were clumped together in a weird way, the scanner would detect a lot of "noise" or jagged spikes.
The paper shows that for dense Sidon sets, the scanner sees a very smooth, flat signal. This smoothness proves that the marbles are spread out evenly across all the different "remainder" groups (like even/odd or mod 3, mod 4, etc.).

Because the signal is so smooth, the author can calculate the total sum of the marbles (or their powers) in any specific group, and the result matches the "expected" average perfectly, with only a tiny, negligible error.

The "Rank-Weighted" Twist

The paper goes a step further. It doesn't just look at the value of the marble (e.g., the number 5); it also looks at the marble's position in the sorted list.

Imagine the marbles are lined up from smallest to largest.
The "rank" is just their position number (1st, 2nd, 3rd...).
The paper calculates sums where the marble's value is multiplied by its position (e.g., $1 \times \text{value}_1 + 2 \times \text{value}_2 + \dots$ ).

The result is the same: even when you weigh the marbles by their position, the total sum in any specific remainder group still matches the perfect mathematical prediction.

The "Almost All" Result

Finally, the paper addresses the "maximal" case—the absolute largest possible Sidon set you can make.

For every single number $n$ , the distribution is very close to perfect.
However, for "almost all" numbers (meaning if you pick a random huge number, it's almost guaranteed to work), the distribution is even tighter.
The paper uses a clever trick involving "prime gaps" (the spaces between prime numbers) to show that for the vast majority of cases, the error in the calculation is incredibly small.

Summary

In plain English, this paper says:

"We used to think that these special number sets were evenly distributed only if we assumed they stayed evenly distributed at the very end. We now know that assumption wasn't necessary. Using a new 'smoothness' detector, we proved that these sets are naturally and perfectly spread out across all number patterns, whether you just count them, add their values, or weigh them by their position. The math works out exactly as the universe intended, without any extra conditions."

The author also notes in the paper that they used an AI tool (OpenAI Codex) to help spot the connection between existing mathematical tools and this specific problem, which led to this new, unconditional proof.

Technical Summary: Power and Rank-Weighted Sums in Dense Finite Sidon Sets

Problem Statement
The paper addresses the distribution of dense finite Sidon sets $S \subset \{1, 2, \dots, n\}$ within arithmetic progressions and residue classes. A set $S$ is Sidon if all pairwise sums $a+b$ (for unordered pairs) are distinct. The central problem is to establish unconditional asymptotic formulae for power sums $\sum_{a \in S} a^\ell$ and rank-weighted sums $\sum_{i} i^s a_i^\ell$ restricted to specific residue classes $a \equiv r \pmod m$ .

Previous work by the author [4] established such asymptotics conditionally, relying on a hypothesis regarding the lower bound of $|S \cap [1, k]|$ over long terminal ranges. As noted by Balasubramanian and Dutta [1], this condition is rarely compatible with the expected distribution of dense Sidon sets. The paper aims to remove this conditional hypothesis to provide an unconditional proof for fixed power sums and their generalizations.

Methodology
The primary methodological shift in this work is the replacement of conditional density assumptions with the Fourier uniformity of extremal Sidon sets, a result established by Ortega and Prendiville [10].

Fourier Uniformity: The proof leverages the bound on the Fourier transform of the indicator function of a Sidon set, specifically $\|\widehat{1_S} - \frac{|S|}{n}\widehat{1_{[1,n]}}\|_\infty \ll \Phi(S, n)$ , where $\Phi(S, n)$ measures the deviation of $|S|$ from the maximal size $n^{1/2}$ .
Abel Summation and Orthogonality: The authors use Fourier orthogonality to estimate the discrepancy of $S$ in finite arithmetic progressions. By combining the Fourier uniformity bound with standard estimates for the Fourier transform of arithmetic progressions, they derive a uniform error term for the count of elements in any progression.
Weighted Summation: For rank-weighted sums (where elements are weighted by their index $i^s$ ), the authors relate the index $i$ to the value $a_i$ via the counting function. Using the discrepancy estimate, they approximate $i \approx \frac{|S|}{n}a_i$ and apply Abel summation to transfer the uniform distribution property to the weighted sums.
Almost-All Results: To improve error terms for "almost all" $n$ , the paper incorporates a large-prime-gap estimate by Heath-Brown [7], utilized previously by Balasubramanian and Dutta [1]. This allows for a sharper bound on the size defect $|S| - n^{1/2}$ for the vast majority of integers $n$ .

Key Contributions and Results

Unconditional Residue-Class Asymptotics (Theorem 1 & Corollary 2): The paper proves that for any finite arithmetic progression $P \subset [1, n]$ , the sum of powers $\sum_{a \in S \cap P} a^\ell$ satisfies:
$\sum_{a \in S \cap P} a^\ell = \frac{|S|}{n} \sum_{a \in P} a^\ell + O(n^\ell \Phi(S, n) \log(2n))$
This result holds unconditionally, removing the previous dependency on specific distribution hypotheses.
Maximal Sidon Sets (Corollary 3): For a maximal Sidon set where $|S| = S_n = n^{1/2} + O(n^{21/80})$ , the paper derives the specific asymptotic for residue classes:
$\sum_{\substack{a \in S \\ a \equiv r \pmod m}} a^\ell = \frac{1}{m(\ell+1)} n^{\ell+1/2} + O(n^{\ell+61/160} \log n)$
This specifically answers the open question posed in the author's previous work [4] regarding the validity of the residue-class formula without extra conditions.
Rank-Weighted Sums (Theorem 4 & Corollary 5): The results are generalized to sums weighted by the rank of the element in the set ( $i^s a_i^\ell$ ). For a Sidon set $S = \{a_1 < \dots < a_t\}$ , the paper establishes:
$\sum_{\substack{1 \le i \le t \\ a_i \in I \\ a_i \equiv r \pmod m}} i^s a_i^\ell = \frac{t^{s+1}}{n^{s+1}} \sum_{\substack{a \in I \\ a \equiv r \pmod m}} a^{s+\ell} + O(t^s n^\ell \Phi(S, n) \log(2n))$
For maximal sets, this yields an error term of $O(n^{\ell+s/2+61/160} \log n)$ .
Almost-All Improvement (Theorem 6): By combining the Fourier uniformity approach with Heath-Brown's prime-gap estimates, the paper shows that for all but $O(N^{4/5+\varepsilon})$ integers $n \le N$ , the error term for maximal Sidon sets can be sharpened to:
$O(n^{\ell+s/2+3/8} \log n)$
This improves upon the error terms found in previous literature for the "almost all" case.

Significance and Author's Claims
The paper claims to provide a "short unconditional proof" of the residue-class power-sum asymptotic for dense Sidon sets. The significance lies in:

Removing Conditional Hypotheses: It resolves the open question from [4] by proving the asymptotic formulae without the restrictive condition on the distribution of $|S \cap [1, k]|$ .
Unification: It demonstrates that both standard power sums and more complex rank-weighted sums are controlled by the same underlying principle: the Fourier uniformity (discrepancy) of the Sidon set on finite arithmetic progressions.
Methodological Clarity: The author notes that while the proofs are not overly difficult given the Fourier uniformity results of Ortega and Prendiville, the connection between these results and the specific problem of weighted sums in residue classes was not immediately obvious.

The author explicitly states that the proofs rely on the Fourier uniformity of extremal Sidon sets (Ortega and Prendiville [10]) and, for the almost-all results, the prime-gap estimates of Heath-Brown [7]. The paper does not propose new applications or future experimental directions but focuses on refining the theoretical understanding of the distribution of Sidon sets. The author also discloses that OpenAI Codex assisted in identifying the applicability of Ortega and Prendiville's results and in formulating the generalizations to rank-weighted sums and almost-all theorems, though the author retains full responsibility for the proofs.