Multiplicative Sidon Sets: Theory and Extremal Bounds

• Multiplicative Sidon sets are collections of positive integers where every unordered pair produces a unique product, with primes serving as a canonical example yet not being extremal.
• Extremal results reveal that maximal subsets in [1,n] achieve sizes of π(n) plus a Θ(n^(3/4)(log n)^(3/2)) correction, achieved by graph-theoretic and combinatorial methods.
• Recent research employs local probabilistic techniques and matching frameworks to derive uniform gap bounds and refined enumeration, advancing both theory and applications.

A multiplicative Sidon set is a set of positive integers in which products of unordered pairs are unique: if $a,b,c,d\in A$ and $ab=cd$, then $\{a,b\}=\{c,d\}$. Equivalently, $A$ contains no non-trivial solution to $ab=cd$, and the product map on unordered pairs is injective. The set of all primes is the basic example, but it is not extremal: classical and recent results show that one can enlarge the primes substantially while preserving the multiplicative Sidon property, and can also impose nontrivial regularity conditions such as small maximal gaps in $[1,n]$ (Doorn et al., 5 Jun 2026).

1. Definition, equivalent formulations, and scope

For a set $A\subseteq \mathbb{N}$, one convenient formalization is

$P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$

together with the product map

$\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$

Then $A$ is multiplicative Sidon precisely when $ab=cd$0 is injective. This formulation makes explicit that squares are part of the constraint: the case $ab=cd$1 is allowed, so $ab=cd$2 must also be uniquely represented (Doorn et al., 3 May 2026).

In multiplicative-group language, used for subsets of $ab=cd$3 or $ab=cd$4, the same property can be expressed as

$ab=cd$5

where the factor $ab=cd$6 accounts for the trivial symmetry $ab=cd$7 and $ab=cd$8. A related ratio formulation uses $ab=cd$9, and both product and ratio viewpoints appear in higher-energy arguments for extracting multiplicative Sidon-type subsets from arbitrary finite sets (Shkredov, 2021).

The terminology is not uniform across the literature. In one strand, a set $\{a,b\}=\{c,d\}$0 is called $\{a,b\}=\{c,d\}$1-multiplicative if $\{a,b\}=\{c,d\}$2 for all $\{a,b\}=\{c,d\}$3; this is a forbidden-ratio problem rather than the classical unique-product condition. In another strand, nonabelian $\{a,b\}=\{c,d\}$4-sets require uniqueness of ordered $\{a,b\}=\{c,d\}$5-fold products in a group. These notions are related in spirit but are not equivalent to the classical multiplicative Sidon property (Wakeham et al., 2011).

2. Extremal size in finite intervals

The classical finite extremal problem asks for the maximal size of a multiplicative Sidon subset of $\{a,b\}=\{c,d\}$6. Erdős proved that maximal multiplicative Sidon subsets of $\{a,b\}=\{c,d\}$7 have size

$\{a,b\}=\{c,d\}$8

where $\{a,b\}=\{c,d\}$9 is the number of primes $A$0. This identifies the prime set as the first-order term and shows that one can add a substantial lower-order population of composites while keeping pairwise products unique (Doorn et al., 5 Jun 2026).

A useful structural picture comes from encoding admissible composites by sparse graphs. If one takes primes $A$1 and forms semiprimes $A$2, then product collisions among such semiprimes correspond to $A$3-configurations in a graph on the prime set. Choosing edges from a $A$4-free graph therefore yields multiplicative Sidon families of semiprimes, and extremal $A$5-free graphs on about $A$6 vertices provide the reservoir that explains the $A$7 correction term (Liu et al., 2018).

This finite-size theorem already rules out a common simplification of the subject: the primes are a canonical multiplicative Sidon set, but they are not maximal in $A$8. The problem is therefore not merely to identify product-unique sets, but to understand how far one can move beyond prime support without introducing a non-trivial equality $A$9.

For higher multiplicative analogues, the asymptotic landscape changes. A multiplicative $ab=cd$0-Sidon set forbids collisions between products of $ab=cd$1 distinct elements. For $ab=cd$2, the maximal size $ab=cd$3 satisfies

$ab=cd$4

improving earlier upper bounds and reflecting a different balance between prime-driven main terms and graph-theoretic error terms (Pach, 2018).

3. Gap problems and the function $ab=cd$5

Beyond cardinality, recent work studies how uniformly multiplicative Sidon sets can be distributed inside $ab=cd$6. The central parameter is

$ab=cd$7

Equivalently, $ab=cd$8 is the least scale at which one can choose a multiplicative Sidon set whose maximal gap in $ab=cd$9 is at most $[1,n]$0 (Doorn et al., 3 May 2026).

Sárközy asked whether one always has $[1,n]$1. This was answered affirmatively: for every $[1,n]$2,

$[1,n]$3

The construction is explicit. Writing $[1,n]$4, one takes

$[1,n]$5

which intersects every interval of length $[1,n]$6 and is multiplicative Sidon by a direct congruence-class argument (Doorn et al., 3 May 2026).

The same paper broke the square-root barrier. With

$[1,n]$7

it proved that for every $[1,n]$8,

$[1,n]$9

This improved on what was previously obtainable from prime gaps alone, namely $A\subseteq \mathbb{N}$0 for all sufficiently large $A\subseteq \mathbb{N}$1, and placed the true order of magnitude between $A\subseteq \mathbb{N}$2 and $A\subseteq \mathbb{N}$3 (Doorn et al., 3 May 2026).

Part II sharpened the exponent substantially. For every $A\subseteq \mathbb{N}$4,

$A\subseteq \mathbb{N}$5

so there exist multiplicative Sidon sets $A\subseteq \mathbb{N}$6 with maximal gap $A\subseteq \mathbb{N}$7. This lowers the known gap exponent from approximately $A\subseteq \mathbb{N}$8 to approximately $A\subseteq \mathbb{N}$9 (Doorn et al., 5 Jun 2026).

These gap results complement the extremal-size theorem rather than compete with it. The size theorem shows that multiplicative Sidon sets can be much denser than the primes; the gap theory shows that one can also force a fairly uniform spatial distribution across $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$0.

4. Mechanisms behind modern gap bounds

Two different proof architectures currently dominate the gap problem. The first, used in Part I, combines primes in short intervals with a matching framework. A key input is the private-prime criterion: if every $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$1 can be written as $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$2, where $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$3 is prime, $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$4, and the primes $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$5 are pairwise distinct across $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$6, then $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$7 is multiplicative Sidon. This reduces product uniqueness to the existence of distinct large prime “labels” attached to the chosen elements. The construction then partitions $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$8 into blocks, seeds early blocks using Baker–Harman–Pintz primes in short intervals, and fills later blocks via a weighted Hall matching based on the Laishram–Murty averaged short-interval prime theorem (Doorn et al., 3 May 2026).

The second architecture, developed in Part II, is local and probabilistic. Its core selection lemma states that if $P_2(A):=\{\{a,b\}:a,b\in A,\ a\le b\},$9 is a family of pairwise disjoint subsets of $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$0 with $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$1, $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$2, and $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$3, then for all sufficiently large $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$4 there exists a multiplicative Sidon set $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$5 with $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$6 for every $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$7. The proof uses the asymmetric Lovász local lemma on bad events coming from non-trivial relations $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$8, together with divisor-function bounds $\mu:P_2(A)\to \mathbb{N},\qquad \mu(\{a,b\})=ab.$9 for $A$0 and large $A$1 (Doorn et al., 5 Jun 2026).

This local-lemma selection already gives a purely probabilistic bound

$A$2

The sharper exponent arises from coupling the local lemma to the distribution of primes in short intervals. The paper defines $A$3 by measuring the exceptional set of $A$4 for which $A$5 contains fewer than $A$6 primes, and proves the reduction

$A$7

A stronger exceptional-set parameter $A$8 is defined by failure of the prime number theorem in $A$9, with the trivial inequality $ab=cd$00 (Doorn et al., 5 Jun 2026).

The final exponent $ab=cd$01 is obtained by importing Gafni–Tao’s bound $ab=cd$02 for every $ab=cd$03. Since $ab=cd$04 in that range, the reduction applies, and letting $ab=cd$05 yields the stated bound. Under the Lindelöf Hypothesis, the same framework gives $ab=cd$06, hence $ab=cd$07 (Doorn et al., 5 Jun 2026).

The constructed sets in Part II have a two-tier form $ab=cd$08. The set $ab=cd$09 consists of large primes, one selected from each “good” interval, while $ab=cd$10 is drawn from “bad” intervals after excluding integers divisible by the chosen primes. The design ensures multiplicative Sidon inside $ab=cd$11 and coprimality between $ab=cd$12 and $ab=cd$13, so cross-collisions are forced to be trivial (Doorn et al., 5 Jun 2026).

5. Infinite multiplicative Sidon sets and enumeration

The finite extremal correction term $ab=cd$14 does not persist uniformly along infinite multiplicative Sidon sets. For an infinite multiplicative Sidon set $ab=cd$15, the relevant scale for the excess over the primes is $ab=cd$16. One theorem states that if

$ab=cd$17

then

$ab=cd$18

A corollary is that every infinite multiplicative Sidon set satisfies

$ab=cd$19

Conversely, there exists a multiplicative Sidon set $ab=cd$20 such that

$ab=cd$21

The lower construction augments the primes by carefully chosen products of four primes from dyadic prime windows, using combinatorial constraints that prevent the relevant product collisions (Pach et al., 2017).

Enumeration yields a different perspective. Let $ab=cd$22 denote the number of multiplicative Sidon subsets of $ab=cd$23. Then

$ab=cd$24

where

$ab=cd$25

The factor $ab=cd$26 counts the independent choices coming from large primes $ab=cd$27, for each of which one may include at most one multiple of $ab=cd$28. Moreover,

$ab=cd$29

in the exponential sense, with the constant $ab=cd$30 arising from

$ab=cd$31

This resolves the enumeration problem initiated by Cameron and Erdős and shows that, although extremal multiplicative Sidon sets have size $ab=cd$32, the total number of such sets is driven by a much richer combinatorial choice structure (Liu et al., 2018).

The same paper extends enumeration to generalized multiplicative $ab=cd$33-Sidon sets. For even $ab=cd$34,

$ab=cd$35

while for odd $ab=cd$36,

$ab=cd$37

for explicitly defined constants $ab=cd$38 based on product-free graphs. In the case $ab=cd$39, the constant $ab=cd$40 satisfies

$ab=cd$41

These asymptotics isolate a graph-theoretic core inside multiplicative collision avoidance (Liu et al., 2018).

6. Generalizations, related frameworks, and open directions

Multiplicative Sidon phenomena extend well beyond subsets of $ab=cd$42. In additive-combinatorial form, higher-energy methods show that any finite subset $ab=cd$43 of the real numbers or of the prime field either contains an additive Sidon-type subset of size $ab=cd$44 or a multiplicative Sidon-type subset of size $ab=cd$45. The mechanism is a dichotomy: either a suitable higher energy is small, in which case random pruning yields Sidon-type structure, or $ab=cd$46 has a highly additive-structured component, on which incidence bounds force multiplicative collisions to be sparse (Shkredov, 2021).

This positive result coexists with sharp obstructions. A construction of Roche-Newton and Warren gives a set $ab=cd$47 such that any subset $ab=cd$48 with $ab=cd$49 is neither an additive nor multiplicative Sidon set. In particular, one cannot expect a universal exponent arbitrarily close to $ab=cd$50 for the larger of the additive and multiplicative Sidon subset sizes (Roche-Newton et al., 2021). A complementary line, phrased in $ab=cd$51 notation, shows that for arbitrary finite sets of integers there are absolute constants $ab=cd$52 and $ab=cd$53 such that the largest additive $ab=cd$54 subset and largest multiplicative $ab=cd$55 subset satisfy

$ab=cd$56

with the case $ab=cd$57 admitting $ab=cd$58 (Jing et al., 2022).

The interaction with additive Sidon structure leads to bi-Sidon problems. A bi-Sidon set is simultaneously additive Sidon and multiplicative Sidon. For every finite $ab=cd$59, the best general lower bound currently cited here is

$ab=cd$60

improving Ruzsa’s earlier $ab=cd$61-scale guarantee (Pach et al., 2024).

Several neighboring notions should be kept distinct. In the forbidden-ratio problem, the maximal density of a $ab=cd$62-multiplicative set is

$ab=cd$63

and the extremal sets are given by even subpowers of $ab=cd$64; this is a dense-ratio-avoidance problem, not the classical unique-product problem (Wakeham et al., 2011). In nonabelian groups, an $ab=cd$65-set requires uniqueness of ordered $ab=cd$66-fold products. For symmetric groups, one has

$ab=cd$67

and the theory connects directly to Cayley digraphs and Turán-type problems (Byrne et al., 9 Sep 2025).

The current open problems are concentrated around sharp exponents and constants. For gaps, the true order of $ab=cd$68 remains open between $ab=cd$69 and $ab=cd$70, and there is no known matching lower bound of the form $ab=cd$71 for any $ab=cd$72 (Doorn et al., 3 May 2026). For infinite multiplicative Sidon sets, the constants $ab=cd$73 and $ab=cd$74 leave a large gap at the $ab=cd$75 scale (Pach et al., 2017). For enumeration, the exact value of the product-free-graph constants $ab=cd$76 and $ab=cd$77 remains unresolved (Liu et al., 2018). These questions suggest that the subject still sits at a point where extremal number theory, prime distribution, sparse graph theory, and probabilistic construction all remain simultaneously decisive.