Sidon Sets: Uniqueness and Applications

• Sidon sets are subsets of groups defined by the uniqueness condition (a+b=c+d implies {a,b} = {c,d}), ensuring distinct additive or multiplicative representations.
• Their study spans additive combinatorics, finite-field geometry, coding theory, and harmonic analysis, leading to optimal constructions and bounds in various settings.
• Modern variants, including q-analogues and operator-space formulations, extend Sidon phenomena to cryptographic functions and network coding applications.

Sidon denotes a cluster of uniqueness and interpolation phenomena that originated with additive $B_2$ sequences and now span additive combinatorics, finite-field geometry, coding theory, harmonic analysis, and operator spaces. In its classical form, a Sidon set is a subset $A$ of an abelian group such that $a+b=c+d$ implies $\{a,b\}=\{c,d\}$; modern variants include multiplicative Sidon sets in $\mathbb N$, Sidon subsets of $\mathbb F_2^t$, Sidon sets in dual groups of compact abelian groups, completely Sidon sets in discrete groups, and the $q$-analogues called Sidon spaces (Cilleruelo et al., 2018, Doorn et al., 3 May 2026, Lewko, 4 Jun 2026, Pisier, 2017, Castello et al., 2023).

1. Core meanings and definitions

The unifying theme is uniqueness of representation under a binary or multilinear operation. In additive combinatorics, a Sidon set $A$ in an abelian group satisfies

$a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$

In $\mathbb F_2^t$, the characteristic-$A$0 formulation is equivalent to requiring that the sum of any four distinct elements is never $A$1. In multiplicative number theory, a multiplicative Sidon set $A$2 satisfies

$A$3

In harmonic analysis on a compact abelian group $A$4 with dual group $A$5, a set $A$6 is Sidon if there exists $A$7 such that

$A$8

for every finite $A$9. In the $a+b=c+d$0-analogue, an $a+b=c+d$1-subspace $a+b=c+d$2 is a Sidon space if for nonzero $a+b=c+d$3,

$a+b=c+d$4

(Cilleruelo et al., 2018, Czerwinski et al., 2024, Lewko, 4 Jun 2026, Doorn et al., 3 May 2026, Castello et al., 2023).

Setting	Defining uniqueness property	Typical reformulation
Additive groups	$a+b=c+d$5	pairwise sums are distinct
$a+b=c+d$6	no four distinct elements sum to $a+b=c+d$7	$a+b=c+d$8
Multiplicative subsets of $a+b=c+d$9	$\{a,b\}=\{c,d\}$0	unordered products are distinct
Dual groups $\{a,b\}=\{c,d\}$1	$\{a,b\}=\{c,d\}$2-$\{a,b\}=\{c,d\}$3 comparability on $\{a,b\}=\{c,d\}$4	bounded interpolation on $\{a,b\}=\{c,d\}$5
Sidon spaces	$\{a,b\}=\{c,d\}$6	$\{a,b\}=\{c,d\}$7

A further abstraction replaces pair sums by a fixed linear form $\{a,b\}=\{c,d\}$8. A set $\{a,b\}=\{c,d\}$9 is then $\mathbb N$0-Sidon if $\mathbb N$1 is injective on $\mathbb N$2. In an infinite vector space, infinite $\mathbb N$3-Sidon sets exist if and only if the coefficient set $\mathbb N$4 has distinct subset sums, and the integer growth bounds become $\mathbb N$5 in general and $\mathbb N$6 for explicit constructions (Nathanson, 2021).

2. Additive combinatorics in integers and finite groups

Classical Sidon theory is organized by the near-square-root bound. For a finite abelian group $\mathbb N$7 of order $\mathbb N$8, a Sidon set $\mathbb N$9 of size $\mathbb F_2^t$0 satisfies $\mathbb F_2^t$1, so dense Sidon means $\mathbb F_2^t$2. A geometric synthesis asserts that all known dense Sidon subsets of abelian groups arise from projective planes through a common construction, and the desarguesian classification recovers the Singer, Bose, Spence, and Hughes–Cilleruelo families; the same framework also yields many examples from nondesarguesian planes and motivates the conjecture that all dense Sidon sets arise in this way (Eberhard et al., 2021).

Several extremal variants are now well developed. For Sidon systems of $\mathbb F_2^t$3-subsets of $\mathbb F_2^t$4, the maximum size $\mathbb F_2^t$5 satisfies

$\mathbb F_2^t$6

with $\mathbb F_2^t$7, $\mathbb F_2^t$8, and $\mathbb F_2^t$9; in the random model $q$0, the threshold for being Sidon is $q$1 (Cilleruelo et al., 2018). For arbitrary $q$2-element subsets of $q$3, if $q$4 denotes the minimum possible size of the largest Sidon subset, then

$q$5

and the same lower bound holds uniformly for $q$6-point subsets of $q$7 (Bailleul et al., 4 May 2026).

The basis problem is a separate line of inquiry. Erdős asked for an infinite Sidon sequence that is an asymptotic basis of order $q$8. Modularly, for all sufficiently large $q$9, there exists a Sidon set in $A$0 that is a basis of order $A$1; in the integers, there exists an infinite $A$2 sequence that is an asymptotic basis of order $A$3, and for every $A$4 there exists a Sidon sequence that is an asymptotic basis of order $A$5 in the sense that every sufficiently large integer is a sum of four elements of the sequence, one of them at most $A$6 (Cilleruelo, 2013). A complementary density result shows that there exists a Sidon set $A$7 with

$A$8

so the triple sumset of a genuine Sidon sequence can already have positive lower density even though the order-$A$9 basis question remains open (Kiss et al., 2021).

3. Sidon sets in $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$0, coding theory, and APN constructions

Over $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$1, Sidon sets are tightly linked to binary linear codes. If $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$2 is represented as the column set of a parity-check matrix $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$3, then the associated code $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$4 satisfies: $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$5 is sum-free iff $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$6, and $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$7 is sum-free Sidon iff $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$8. Consequently, the nonexistence of a binary $a+b=c+d \Rightarrow \{a,b\}=\{c,d\}.$9 code is equivalent to the upper bound $\mathbb F_2^t$0 for Sidon sets in $\mathbb F_2^t$1. The trivial estimate

$\mathbb F_2^t$2

is sharpened to $\mathbb F_2^t$3 for odd $\mathbb F_2^t$4, and to an explicit even-$\mathbb F_2^t$5 bound of the form

$\mathbb F_2^t$6

for $\mathbb F_2^t$7. The same paper classifies maximal Sidon sets up to affine equivalence for $\mathbb F_2^t$8, proves that every maximal Sidon set in $\mathbb F_2^t$9 has size $A$00, and shows that the possible sizes in $A$01 are $A$02 (Czerwinski et al., 2023).

A second major theme identifies Sidon sets with graphs of cryptographically strong functions. For $A$03, the graph

$A$04

is Sidon if and only if $A$05 is APN. Intersecting $A$06 with a hyperplane corresponding to a maximizing Walsh coefficient produces a Sidon set in $A$07 of size

$A$08

Using 8-bit APN functions of linearity $A$09, this yields Sidon sets of size $A$10 in $A$11, improving the previous best-known size $A$12, and it produces a binary $A$13 code. The same analysis proves the general APN bound $A$14 (Czerwinski et al., 2024).

The cryptographic analogy has recently been pushed further. For Sidon sets $A$15, minimal linearity is characterized exactly by the $A$16-cover property: every $A$17 has exactly $A$18 representations as $A$19 with $A$20. This classification is recast through the Cayley graph of the Boolean function $A$21, yielding strongly regular graphs; in particular, the unique rank-$A$22 strongly regular graph with parameters $A$23 is realized as such a Cayley graph, and the best-known lower bound for the largest Sidon set in $A$24 is improved by $A$25 for all $A$26 (Thornburgh, 19 Jan 2025). Related work on polynomial graphs in $A$27 shows that planar polynomials make $A$28 a maximum Sidon set, while various monomial and cubic candidates over $A$29 are excluded by exact multiplicity criteria (Afifurrahman et al., 2021).

4. Harmonic-analytic, probabilistic, and operator-space formulations

In harmonic analysis, Sidon sets are interpolation sets for Fourier coefficients. For a compact abelian group $A$30 with dual $A$31, a subset $A$32 is Sidon precisely when bounded data on $A$33 extend to Fourier transforms of bounded continuous functions or bounded measures with uniform control. In dual groups of bounded torsion, this interpolation property is equivalent to a purely combinatorial decomposition statement: $A$34 The final missing prime-power case is proved by combining Pisier’s proportional quasi-independence theorem, Bourgain’s projection theorem, a local dimension bound for reduced signed relations, and a Rado–Horn support-partition argument; in prime-power groups, the number of quasi-independent pieces is bounded by

$A$35

(Lewko, 4 Jun 2026).

For torsion-free dual groups, Sidon sets admit an even sharper proportional structure. A Sidon set is proportionally $A$36-degree independent for every $A$37, and for every $A$38 one can extract proportionally large subsets with Sidon constant at most $A$39, the minimum possible value. The argument combines Pisier’s proportional characterization, higher-order independence, and Riesz-product interpolation; in torsion groups such uniform small constants can fail (Hare et al., 2018).

The same vocabulary extends beyond characters. Any uniformly bounded $A$40-orthonormal system is $A$41-fold tensor Sidon, improving Bourgain–Lewko’s $A$42-fold result, and a bounded system is randomly Sidon if and only if it is $A$43-fold tensor Sidon, equivalently $A$44-fold tensor Sidon for some or all $A$45. Pisier’s proof uses Talagrand’s majorizing measure theorem, Gaussian domination, and a tensor decomposition

$A$46

(Pisier, 2016).

A noncommutative version replaces classical Banach-space Sidonicity by operator-space structure. A subset $A$47 of a discrete group $A$48 is completely Sidon if its span in $A$49 is completely isomorphic to $A$50 with maximal operator space structure. Completely Sidon sets are stable under finite unions, bounded Hermitian functions on symmetric completely Sidon sets extend to positive definite functions, and $A$51 is completely Sidon if and only if the dual operator space $A$52 is exact (Pisier, 2017).

5. Sidon spaces and $A$53-analogues

Sidon spaces are the $A$54-analogues of Sidon sets. An $A$55-subspace $A$56 is Sidon exactly when

$A$57

for nonzero $A$58, or equivalently when

$A$59

Roth, Raviv, and Tamo showed that this is equivalent to saying that the cyclic orbit $A$60 has full size $A$61 and minimum subspace distance $A$62, making Sidon spaces precisely the optimal one-orbit cyclic subspace codes of that type (Castello et al., 2023).

A large class of constructions lives inside sums of two multiplicative cosets of a fixed subfield. Writing $A$63, one considers

$A$64

or, in graph form,

$A$65

for a linearized polynomial $A$66. The paper “Constructions and equivalence of Sidon spaces” characterizes when $A$67 is Sidon through an intrinsic “Sidon space property” for $A$68, proves that scattered polynomials yield Sidon space polynomials, establishes a sharp monomial criterion in degree-$A$69 extensions, gives a direct-sum construction, and formulates semilinear equivalence in terms of $A$70-actions and $A$71-orbits (Castello et al., 2023).

The generalized theory replaces pairwise multiplicative uniqueness by $A$72-fold multiplicative uniqueness. An $A$73-Sidon space is a subspace $A$74 such that equality of two $A$75-fold products of nonzero elements forces equality of the multisets of one-dimensional $A$76-subspaces. If $A$77 denotes the $A$78-span, then

$A$79

and equality implies that $A$80 is $A$81-Sidon. The same framework gives lower bounds on $A$82, prime-degree strengthenings, constructions from $A$83-sets and irreducible polynomials, and a scattered-polynomial construction

$A$84

which is $A$85-Sidon whenever $A$86 is scattered and $A$87 generates an extension of degree at least $A$88 (Castello, 2023). These spaces are motivated not only by combinatorics but also by random linear network coding, because one-orbit cyclic subspace codes derived from Sidon spaces achieve large size together with minimum distance $A$89 (Castello, 2023).

6. Multiplicative, geometric, and current developments

The multiplicative analogue asks for subsets of $A$90 with unique unordered products. If $A$91 denotes the least $A$92 such that some multiplicative Sidon set $A$93 meets every interval $A$94, then primes show that multiplicative Sidon sets can be large, while Erdős proved that the maximum size is

$A$95

Recent work answers Sárközy’s square-root question by proving

$A$96

via the explicit congruence class construction $A$97 with $A$98, and then improves this asymptotically to

$A$99

using primes in short intervals, a private-prime Sidon criterion, and a weighted Hall matching argument. A notable auxiliary feature is that the square-root proof was autonomously discovered and formally verified in Lean by Aristotle, and the power-saving argument was formalized in Lean under its analytic inputs (Doorn et al., 3 May 2026).

A subsequent refinement lowers the exponent dramatically: $a+b=c+d$00 The proof introduces a Lovász Local Lemma selection principle for choosing one element from each of many large local sets while preserving multiplicative Sidonicity, then combines it with exceptional-set bounds for primes in short intervals. Under Lindelöf- or RH-type estimates one further gets the conditional bound $a+b=c+d$01 (Doorn et al., 5 Jun 2026). This suggests that the exact order of $a+b=c+d$02, currently bracketed between logarithmic lower bounds and subpower upper bounds, remains one of the central open problems in the multiplicative theory.

Sidon sequences also admit a geometric incarnation in CAT(0) geometry. Given a finite Sidon sequence $a+b=c+d$03, Barré and Pichot construct link graphs $a+b=c+d$04 such that

$a+b=c+d$05

With equilateral triangles, this is exactly the nonpositive-curvature link condition. The same paper shows that the number of representations of an integer as an alternating triple sum $a+b=c+d$06 determines the local ring data governing embedded Euclidean flats in the associated CAT(0) complex (Barré et al., 2023).

Taken together, these developments show that Sidon phenomena are not a single theorem but a recurrent structural principle. Depending on the ambient category, Sidon means unique additive representation, unique multiplicative representation, bounded Fourier interpolation, operator-space $a+b=c+d$07-structure, or uniqueness of multiplicative factorizations in subspace codes. A plausible implication is that progress in one setting—Walsh-spectral methods over $a+b=c+d$08, projection and quasi-independence in harmonic analysis, or prime-distribution estimates in multiplicative problems—often exports nontrivially to the others (Thornburgh, 19 Jan 2025, Lewko, 4 Jun 2026, Doorn et al., 5 Jun 2026).