Sidon Set: Structure & Applications

• Sidon sets are defined as collections of integers (or group elements) with unique sum representations, ensuring a + b = c + d only when the unordered pairs coincide.
• Key constructions include Bose–Chowla, Ruzsa's probabilistic methods, and geometric approaches linked to finite projective planes, each refining density bounds.
• Sidon sets underpin applications in coding theory, cryptography and harmonic analysis, with recent research sharpening asymptotic growth rates and extremal parameters.

A Sidon set is a set of integers (or, more generally, elements of an abelian group) with the strong additive uniqueness property: all solutions of the equation $a + b = c + d$ with $a,b,c,d$ in the set are necessarily trivial, meaning the unordered pairs $\{a,b\}$ and $\{c,d\}$ coincide. Sidon sets, also known as $B_2$ sets or Golomb rulers, are a foundational concept in additive combinatorics, bridging number theory, harmonic analysis, coding theory, geometry, and algebraic design.

1. Definitions, Variants, and Core Properties

A set $A$ of (typically) positive integers is a Sidon set if for every $n$ there is at most one solution (up to ordering) to $a + b = n$ with $a, b \in A$, that is,

$$R_{2,A}(n) = \#\{(a,b) \in A \times A : a + b = n,\, a \leq b\} \leq 1 \qquad \forall\, n \in \mathbb{N}.$$

Equivalently, for all quadruples $a, b, c, d \in A$, $a + b = c + d$ implies $\{a, b\} = \{c, d\}$.

Generalizations include $B_h[g]$ sets (where $h \geq 2$ and $g \geq 1$): here, each $n$ is allowed at most $g$ (unordered) $h$-term representations. The classical Sidon set is the $B_2[1]$ case. A Sidon set in a finite abelian group $G$ is a subset such that every nonzero $d \in G$ can be expressed as $a - b$ (or $a+b = c+d$) in at most one way with $a, b, c, d \in S$ unless the quadruple is trivial.

Key parameters and concepts:

• Maximal size $F_2(n)$: largest Sidon subset of $\{1,2,\dots,n\}$
• Diameter: $diam(A) = \max(A) - \min(A)$ for $A$ a finite Sidon set
• Reciprocal sum: $\sum_{a\in S} 1/a$ for infinite $S$, with supremum known as the distinct distance constant (DDC)
• Set systems ("Sidon systems"): families of subsets with pairwise-distinct sumsets

Quantitative bounds:

• Classical results provide $F_2(n) = n^{1/2} + O(n^{1/4})$, with recent improvements to secondary constants (O'Bryant, 2022, Carter et al., 2023).
• Infinite Sidon sets with density $\gg x^{\sqrt{2}-1+o(1)}$ are constructed (Maldonado, 2011).

2. Extremal Size, Structural Results, and Density

The asymptotic extremal size of finite Sidon sets is pinned down by both upper and lower bounds. The largest Sidon subset $F_2(n)$ of $\{1,2,\dots,n\}$ satisfies

$F_2(n) \le n^{1/2} + c\, n^{1/4},$

with $c \approx 0.99703$ (O'Bryant, 2022) and further refined to $c \approx 0.98183$ (Carter et al., 2023).

Lower bounds arise from explicit constructions:

• Bose–Chowla construction: using finite fields, gives $F_2(n) \ge (1 - o(1)) n^{1/2}$ (Nathanson, 2021).
• Ruzsa's probabilistic construction: there exists an infinite $S \subset \mathbb{N}$ with $| S \cap [1,x] | \gg x^{\sqrt{2}-1}$ (Maldonado, 2011).
• Greedy algorithm: stepwise inclusion yields $|S \cap [1,n]| \sim n^{1/3}$, strictly sparser than the previous methods.

In abelian groups of order $n$, maximal Sidon sets have size at most $(1+o(1)) n^{1/2}$, and all known dense examples arise from projective planes (Eberhard et al., 2021). Constructions using BCH codes in $\mathbb{Z}_2^n$ yield maximal Sidon sets of size $O((n 2^n)^{1/3})$ (Redman et al., 2021).

3. Algebraic, Geometric, and Combinatorial Constructions

The construction of large Sidon sets is intertwined with the structure of algebraic curves, difference sets, and projective/hyperplane geometries:

Construction type	Host structure	Size of Sidon set
Bose–Chowla	$\mathbb{F}_{q^h}$ via polynomials	$q$ in $[1, q^h-2]$ for order $h$
Erdős–Turán (parabola)	$G=K^2$, $S = \{(x, x^2) : x \in K\}$	$|K|$ in $G$
BCH code-derived sets	$\mathbb{Z}_2^n$	size $O((n2^n)^{1/3})$ (maximal)
Projective plane stabilizer	Abelian group acting transitively	$\sim n^{1/2}$ if $|G| = n$

The relationship with incidence geometry is evident: all known dense Sidon sets stem from structures corresponding to projective planes; for abelian $G$ acting on a projective plane $P$, $S = \{ g : p^g \in \ell \}$ is Sidon in $G$ (Eberhard et al., 2021). Conversely, it is conjectured that every dense Sidon set arises in this manner.

Unique factorization in algebraic number fields (e.g., through arguments of Gaussian primes) is a central tool in probabilistic constructions (Maldonado, 2011)—"digit block rearrangement" preserves additive uniqueness and allows efficient translation from multiplicative to additive structures.

4. Growth, Asymptotics, and Reciprocals

For a Sidon set $S \subset \mathbb{N}$ of size $k$, recent work sharpens the minimum diameter:

$$diam(S) \ge k^2 - b k^{3/2} - O(k), \quad b < 1.96365$$

(Carter et al., 2023). Equivalently, for any large $n$, the largest Sidon set in $\{1,2,\ldots, n\}$ has cardinality at most $n^{1/2} + 0.98183 n^{1/4} + O(1)$.

Infinite Sidon sequences with prescribed densities and growth rates have also been investigated. For linear forms, necessary and sufficient conditions for the existence of infinite $\varphi$-Sidon sets are that the set of coefficients has distinct subset sums (Nathanson, 2021). Growth bounds in these generalized settings match classical exponents: $|A \cap [1, n]| \ll n^{1/h}$ for $B_h[1]$ sets.

On the analytic side, the reciprocal sum of infinite Sidon sets, known as the distinct distance constant (DDC), is

$$\mathrm{DDC} = \sup \left\{ \sum_{s \in S} \frac{1}{s} : S \text{ is a Sidon set} \right\}.$$

It was recently shown, via compactness of the set of Sidon generating functions and continuity of integrals, that the supremum is attained (Riblet et al., 27 May 2025), and the best upper bound is improved to $2.247307$.

5. Higher-Order Sidon Sets and Set Systems

Generalizations to set systems and higher-order sumsets have been central in recent work:

• Sidon systems: families $\mathcal{A}$ of $k$-subsets of $[n]$ so that $A+B$ with $A,B \in \mathcal{A}$ are pairwise distinct (Cilleruelo et al., 2018, Wötzel, 2022).
  • Bounds: $F_k(n) \leq \binom{n-1}{k-1} + n - k$, with $F_k(n) = (1-o(1))\binom{n}{k-1}$ for large $n$ (Wötzel, 2022).
• Random Sidon systems: threshold for a random family of $k$-subsets to be a Sidon system is at $p \sim n^{-(2k+1)/4}$ (Cilleruelo et al., 2018).
• $B_h$-Sidon sets: all $h$-term sums are unique; the Bose–Chowla argument via finite fields continues to be fundamental (Nathanson, 2021).
• Sidon-type conditions in combinatorial designs: set systems (e.g., t-adesigns) require every $t$-subset to appear with unique frequencies—a direct translation of additive uniqueness (Dukes et al., 2012).

6. Applications in Harmonic Analysis, Coding, and Cryptography

Sidon sets play a central role in harmonic analysis, where they provide sets of characters with quasi-independence and are crucial in the theory of Sidon spaces and interpolation (Hare et al., 2018). In coding theory, Sidon sets underpin the construction of error-correcting codes and difference sets, with connections made precise in the context of $A_n$ lattices (Kovačević, 2014).

In cryptography, Sidon sets correspond to the graphs of almost perfect nonlinear (APN) functions. Recent work has generalized cryptographic results about APN functions and their Walsh transforms to arbitrary Sidon sets in $\mathbb{F}_2^n$, identifying when the Cayley graph of appropriate Boolean functions is strongly regular and using this to classify and construct $k$-covers and associated combinatorial structures (Thornburgh, 19 Jan 2025).

In the context of the ElGamal cryptosystem, Sidon set structure ensures equidistribution properties for the graph of the discrete log map, with implications for pseudorandomness and cryptographic unpredictability (Niehues et al., 2017).

7. Methodological and Analytical Techniques

Sidon sets have been constructed and analyzed via a variety of sophisticated methodologies:

• Probabilistic methods: random selection with deletion of "bad" elements (to avoid sum collisions), coupled with concentration inequalities (e.g., Kim–Vu polynomial concentration) for sumsets (Kiss et al., 2021).
• Analytic/compactness arguments: using topological properties of generating functions and continuous functionals to guarantee the existence of extremal Sidon sets for functionals such as the reciprocal sum (Riblet et al., 27 May 2025).
• Fourier/Walsh analysis: generalizations of cryptographic results to arbitrary Sidon sets, equating minimal linearity and strong regularity in associated Cayley graphs (Thornburgh, 19 Jan 2025).
• Numerical optimization: sharp upper bounds for $B_2[g]$ sets have been achieved by optimizing over auxiliary functions subject to analytic constraints (Habsieger et al., 2016).
• Geometric constructions: lattice packing and tiling arguments in the geometry of numbers clarify the connection between Sidon sets and perfect codes (Kovačević, 2014).
• Combinatorial variations ('Sidon systems'): extremal combinatorics for set systems and higher-order sumsets (Wötzel, 2022, Cilleruelo et al., 2018).

Conclusion

The theory of Sidon sets is a rich intersection of additive combinatorics, geometry, harmonic analysis, finite group theory, and applications such as coding theory and cryptography. The core Sidon property—uniqueness of pairwise sums—drives a wide array of extremal, structural, and probabilistic questions, with modern research refining classical bounds, uncovering new algebraic structures, and generalizing to broader combinatorial and analytic frameworks. Advances include the exact existence of optimal Sidon sets for important functionals, sharp asymptotic formulas and error terms for their density and support, deep geometric interpretations, and connections to strong spectral and pseudorandomness properties in applied mathematics and information theory.