Power and rank-weighted sums in dense finite Sidon sets

Abstract: Let $S\subset {1,2,\ldots,n}$ be a Sidon set with $|S|=n<sup>{1/2}+O(n<sup>{1/2-δ})$ for some fixed $δ&gt;0$. This article provides the following expected asymptotic formula $$\sum_{\substack{a\in S\ a\equiv r\pmod{m}}} a^\ell =\frac{1}{m(\ell+1)}n^{\ell+1/2} +o\left(n^{{\ell+1/2}\right),}$$ where $m\geq 1$, $0\leq r&lt;m$, and $\ell\geq 0$ are three integers. This removes the additional hypothesis in a previous residue-class asymptotic formula by the author. The proof uses the Fourier uniformity of extremal Sidon sets due to Ortega and Prendiville.

• The paper introduces unconditional asymptotic formulae for power and rank-weighted sums over dense Sidon sets, providing sharp main terms and improved error bounds.
• It leverages refined Fourier-analytic techniques to control discrepancies in arithmetic progressions and residue classes, enhancing classical approaches.
• The findings impact additive number theory and related applications like pseudorandomness and combinatorial designs, suggesting avenues for generalizing to higher-order structures.

Asymptotics for Power and Rank-weighted Sums in Dense Finite Sidon Sets

Introduction

This work establishes new unconditional asymptotic formulae for power and rank-weighted sums over Sidon subsets $S \subset \{1,2,\dots,n\}$ of nearly maximal size. Classical analysis of finite Sidon sets, with $|S| \sim n^{1/2}$, focused on the distribution of elements and their sumsets. However, precise enumerative statistics such as the sum of $a^\ell$ for $a \in S$ and their behavior in residue classes have historically been subject to technical conditionality or limited error bounds. This paper eliminates these defects for a broad class of weighted sums, answering open questions in the literature and refining the understanding of Sidon set statistics using modern Fourier-analytic tools.

Main Results

The central contributions are sharp asymptotic formulae for sums over dense Sidon sets that are either power-weighted, supported in arithmetic progressions, or rank-weighted, providing precise main terms and nontrivial bounds on the error terms. The key results can be summarized as follows:

• For $S$ a Sidon set in $[1,n]$ of size $|S| = n^{1/2} + O(n^{1/2-\delta})$, and for all fixed $m \geq 1$, $0 \leq r < m$, and $\ell \geq 0$,

$|S| \sim n^{1/2}$0

holds uniformly, unconditionally, and with effective error terms.

• For the rank-weighted sum,

$|S| \sim n^{1/2}$1

a precise asymptotic of the shape $|S| \sim n^{1/2}$2 an explicit error term is established; this extends earlier conditional results, sharpens error bounds, and generalizes to simultaneous residue and rank constraints.

• The error terms are materially improved using Fourier-uniformity: for Sidon sets $|S| \sim n^{1/2}$3 with $|S| \sim n^{1/2}$4, the discrepancy of $|S| \sim n^{1/2}$5 in arithmetic progressions is bounded uniformly by $|S| \sim n^{1/2}$6, matching the true density up to a small secondary term.
• The almost-all $|S| \sim n^{1/2}$7 version yields $|S| \sim n^{1/2}$8 secondary terms (in the exponent) outside an exceptional set of $|S| \sim n^{1/2}$9 of size $a^\ell$0, via analysis of maximal Sidon sets and prime gap bounds.

These results fundamentally rest on the recent quantitative Fourier-uniformity theorem for extremal Sidon sets [Ortega, Prendiville 2023], which allows transfer of global equidistribution to local arithmetic progressions and residue constraints.

Technical Innovations

The analysis is based on refined Fourier-analytic bounds controlling the discrepancy of Sidon sets in intervals and arithmetic progressions. The key technical device is that for any Sidon set $a^\ell$1, the difference $a^\ell$2 is bounded by $a^\ell$3, delivering robust uniformity over nontrivial Fourier modes.

By combining this Fourier control with Abel summation and elementary symmetric function arguments, one can precisely transfer the uniformity error into main-term and error-term balances for weighted sums, even when structure such as residue constraints or rank weights are imposed. In particular, the error analysis reveals that the main contribution to discrepancies in such sums arises not from the choice of residue or weight, but from the defect in both Sidon set cardinality and the inherent Fourier bias in $a^\ell$4.

Moreover, the work leverages accurate estimates of Sidon set sizes for almost all $a^\ell$5 based on prime gap theorems (notably Heath-Brown's), allowing for further error improvements in the almost-all sense.

Numerical Results and Claims

The established asymptotic formulae provide explicit main terms and secondary terms. For example, for $a^\ell$6 maximal,

$a^\ell$7

with the implied constant depending only on $a^\ell$8. For the general power/rank-weighted case,

$a^\ell$9

uniformly, removing conditionality and auxiliary hypotheses present in prior results.

A strong claim of this paper is that the main asymptotic holds unconditionally for every fixed residue class, in contrast with previous work which required density conditions or only supplied results for almost all $a \in S$0.

Implications and Future Directions

These results settle the precise leading order and next-order behavior for sums of powers and related statistics in dense finite Sidon sets, both globally and in local arithmetic progressions. From a number-theoretic perspective, they clarify the regularity and uniformity properties of extremal additive sets, confirming that dense Sidon sets closely mimic random sets in their power-sum statistics, both unweighted and with additional arithmetic or rank restrictions.

Practically, these findings have consequences for pseudorandomness constructions, combinatorial designs, and error-correcting codes where Sidon sets play a role. The methodology—exploiting strong Fourier-uniformity—could be extended to other additive combinatorial structures, and any conceivable improvement in quantitative Fourier-uniformity bounds for such sets would have immediate downstream influence on error terms for related statistics.

Future work might examine higher-order analogues (e.g., $a \in S$1 sets), connections to set addition spectra, or pursue sharper uniformity bounds via further advances in exponential sum estimates. Understanding the limits of Sidon set statistics in non-standard settings or under additional algebraic structure remains a fertile area for research.

Conclusion

The paper delivers a substantial advancement in the analysis of dense finite Sidon sets, establishing unconditional, uniform asymptotics for power sums and rank-weighted sums, with improved error bounds and applicability to local and residue-class analysis. Use of Fourier-analytic techniques as applied to extremal additive combinatorics sets a template for further exploration of equidistribution phenomena in structured sets, suggesting new pathways for both theoretical exploration and practical application in additive number theory and beyond.

Reference:

"Power and rank-weighted sums in dense finite Sidon sets" (2606.15041)