Multiplicative Sidon Sets

• Multiplicative Sidon sets are subsets of natural numbers where every unordered pair of elements produces a unique product, with primes serving as a primary example.
• Research focuses on determining the maximum size of these sets in [n] and utilizes graph-theoretic and random-block methods to construct sets with refined asymptotic bounds.
• Recent advances address interval-distribution and gap problems by applying prime-distribution estimates and combinatorial techniques to improve exponent bounds.

A multiplicative Sidon set is a set $S\subseteq \mathbb N$, or more generally $S\subseteq [n]=\{1,2,\dots,n\}$, such that the equation $ab=cd$ with $a,b,c,d\in S$ has only the trivial solutions for which $\{a,b\}=\{c,d\}$ as multisets. Equivalently, all pairwise products of elements of $S$ are distinct up to order. The topic combines extremal number theory, graph-theoretic encoding of multiplicative collisions, and analytic information on primes, and it now includes both classical size questions and recent interval-distribution problems concerning how small the maximal gaps in such sets can be (Doorn et al., 5 Jun 2026, Liu et al., 2018).

1. Definition and equivalent formulations

In its classical form, the multiplicative Sidon property requires uniqueness of unordered factorizations inside the set: if $a,b,c,d\in S$ and $ab=cd$, then necessarily $\{a,b\}=\{c,d\}$. The same definition appears in finite and infinite settings, and the primes furnish the basic example, since unique factorization implies that distinct unordered pairs of primes do not share a product (Doorn et al., 5 Jun 2026).

Several equivalent formulations are standard. In multiplicative-group language, $S$ is multiplicative Sidon if every non-unit quotient has a unique representation: if $S\subseteq [n]=\{1,2,\dots,n\}$0 with $S\subseteq [n]=\{1,2,\dots,n\}$1, then $S\subseteq [n]=\{1,2,\dots,n\}$2 (Shkredov, 2021). For a finite set $S\subseteq [n]=\{1,2,\dots,n\}$3, one can also express the condition through multiplicative energy,

$S\subseteq [n]=\{1,2,\dots,n\}$4

Then $S\subseteq [n]=\{1,2,\dots,n\}$5 is multiplicative Sidon if and only if $S\subseteq [n]=\{1,2,\dots,n\}$6, since the only admissible quadruples are the two trivial reorderings of each pair (Roche-Newton et al., 2021).

This notion is the multiplicative analogue of an additive Sidon set, where one requires uniqueness of sums or differences rather than products. The analogy is structurally useful but not exact: multiplicative problems interact much more directly with prime factorization, divisor bounds, and prime-distribution estimates (Shkredov, 2021).

2. Extremal size in $S\subseteq [n]=\{1,2,\dots,n\}$7 and enumeration

For subsets of $S\subseteq [n]=\{1,2,\dots,n\}$8, the first question is the maximum possible cardinality of a multiplicative Sidon subset. Erdős showed that the largest multiplicative Sidon subset of $S\subseteq [n]=\{1,2,\dots,n\}$9 has size

$ab=cd$0

so the primes determine the main term, and the room for improvement above $ab=cd$1 is of lower order (Doorn et al., 5 Jun 2026). This result established that the extremal problem is not about changing the leading asymptotic scale, but about exploiting a comparatively thin family of additional composite elements.

The lower-bound constructions and later structural analyses are closely tied to auxiliary sparse graphs. One classical description is that one selects one element from each large prime $ab=cd$2 and then adds further composite products coming from a suitable $ab=cd$3-free graph on small primes (Liu et al., 2018). This graph-theoretic encoding is pervasive in the subject: a multiplicative collision $ab=cd$4 corresponds to a short cycle in a carefully chosen factor graph, and extremal graph bounds then control how many elements may be added without violating the Sidon property.

The enumeration problem was resolved by Liu and Pach. Writing $ab=cd$5 for the number of multiplicative Sidon subsets of $ab=cd$6, they showed that

$ab=cd$7

for a function

$ab=cd$8

and $ab=cd$9 (Liu et al., 2018). The factor $a,b,c,d\in S$0 reflects the dominant freedom coming from large primes: for each $a,b,c,d\in S$1, one may choose at most one multiple of $a,b,c,d\in S$2, or choose none.

A recurring misconception is to identify the extremal-size problem with the interval-distribution problem. The former asks how many elements a multiplicative Sidon set in $a,b,c,d\in S$3 can contain, while the latter asks how evenly those elements can be placed. The primes are near-optimal for size in the sense above, but they are not automatically near-optimal for gap size.

3. Gap problems and interval distribution

Recent work has introduced a different parameter. One form, denoted $a,b,c,d\in S$4, is the infimum of all $a,b,c,d\in S$5 such that there exists a multiplicative Sidon set $a,b,c,d\in S$6 meeting every interval $a,b,c,d\in S$7 (Doorn et al., 3 May 2026). Another paper uses $a,b,c,d\in S$8 for the same type of quantity, namely the smallest $a,b,c,d\in S$9 such that some multiplicative Sidon subset of $\{a,b\}=\{c,d\}$0 meets every interval of length $\{a,b\}=\{c,d\}$1 (Doorn et al., 5 Jun 2026). This notational variation coexists with the older use of $\{a,b\}=\{c,d\}$2 for maximal size in other parts of the literature.

Sárközy asked whether one always has $\{a,b\}=\{c,d\}$3. Van Doorn, Monticone and Tang answered this affirmatively, proving

$\{a,b\}=\{c,d\}$4

for every integer $\{a,b\}=\{c,d\}$5 (Doorn et al., 3 May 2026). Their elementary construction takes $\{a,b\}=\{c,d\}$6 and

$\{a,b\}=\{c,d\}$7

which hits every interval of length $\{a,b\}=\{c,d\}$8; the main content is then to verify that this congruence class is multiplicative Sidon. The same paper reports that this proof was autonomously discovered and formally verified in Lean by Aristotle (Doorn et al., 3 May 2026).

The same 2026 paper then crossed the $\{a,b\}=\{c,d\}$9-barrier by proving

$S$0

using a bipartite-matching scheme built from short-interval prime information and a weighted Hall-lemma argument (Doorn et al., 3 May 2026). Shortly afterward, “Gaps in Multiplicative Sidon Sets II” improved the exponent further to

$S$1

that is, $S$2 (Doorn et al., 5 Jun 2026).

The proof of the improved bound has two stages. First, a random-block construction combined with the asymmetric Lovász Local Lemma yields the benchmark exponent $S$3: if disjoint sets $S$4 satisfy $S$5, the number of sets is at most $S$6, and $S$7, then one can choose one element $S$8 so that $S$9 is multiplicative Sidon (Doorn et al., 5 Jun 2026). Second, the argument introduces an exponent $a,b,c,d\in S$0 measuring the total Lebesgue measure of points $a,b,c,d\in S$1 for which the interval $a,b,c,d\in S$2 contains fewer than $a,b,c,d\in S$3 primes. The key implication is: $a,b,c,d\in S$4 One partitions $a,b,c,d\in S$5 into blocks of length $a,b,c,d\in S$6, labels blocks as good or bad depending on whether they contain enough primes exceeding $a,b,c,d\in S$7, chooses one prime from each good block, and then applies the random multiplicative-Sidon lemma only to the bad blocks after shielding them from multiplicative collisions by the previously chosen primes (Doorn et al., 5 Jun 2026).

The final exponent $a,b,c,d\in S$8 comes from inserting the latest available prime-distribution input into this framework. In particular, the paper states that the work of Gafni–Tao shows that for all $a,b,c,d\in S$9, one has $ab=cd$0, which yields the final bound (Doorn et al., 5 Jun 2026). This suggests that further progress on gaps is tightly coupled to sharper control of prime-poor intervals.

4. Infinite multiplicative Sidon sets

For an infinite multiplicative Sidon set $ab=cd$1, the natural counting function is

$ab=cd$2

The question is then how much larger $ab=cd$3 can be than $ab=cd$4 along an infinite sequence of scales. The 2017 paper “On infinite multiplicative Sidon sets” proves complementary upper and lower results and states that the correct gauge for the second-order term is

$ab=cd$5

up to absolute constants (Pach et al., 2017).

On the constructive side, the paper builds an infinite multiplicative Sidon set $ab=cd$6 for which

$ab=cd$7

for all large $ab=cd$8 and some positive constant $ab=cd$9 (Pach et al., 2017). The construction proceeds blockwise. For each $\{a,b\}=\{c,d\}$0, let $\{a,b\}=\{c,d\}$1 be the primes in $\{a,b\}=\{c,d\}$2; from each $\{a,b\}=\{c,d\}$3, a combinatorial lemma produces a family $\{a,b\}=\{c,d\}$4 of four-element subsets satisfying intersection and four-pair avoidance properties, with $\{a,b\}=\{c,d\}$5. One then sets

$\{a,b\}=\{c,d\}$6

and checks from the combinatorial properties of the $\{a,b\}=\{c,d\}$7 that $\{a,b\}=\{c,d\}$8 is multiplicative Sidon (Pach et al., 2017).

The upper bound is more intricate. The argument decomposes $\{a,b\}=\{c,d\}$9 according to the smallest factor in representations $S$0, uses $S$1-free graph estimates to control certain subclasses, and then handles the remaining elements by further factorization into numbers with many prime factors in prescribed windows (Pach et al., 2017). The resulting picture differs sharply from the finite extremal problem in $S$2, where the excess over the primes is of order $S$3. This suggests that maintaining the multiplicative Sidon property simultaneously at all scales is substantially more restrictive than optimizing at a single scale.

5. Variants, generalized notions, and notation

The term “multiplicative Sidon” appears in several related but distinct senses.

A multiplicative $S$4-Sidon set requires that the equation

$S$5

have no solution with all $S$6 factors distinct. For $S$7, Pach proved that the largest multiplicative $S$8-Sidon subset of $S$9 satisfies

$S\subseteq [n]=\{1,2,\dots,n\}$00

improving the previously known best upper bound (Pach, 2018). The proof builds a hexagon-free graph encoding triple-product collisions and combines $S\subseteq [n]=\{1,2,\dots,n\}$01-extremal bounds with analytic estimates on integers having prescribed numbers of prime factors (Pach, 2018).

A broader family is given by multiplicative $S\subseteq [n]=\{1,2,\dots,n\}$02-sets. Here one requires that for every integer $S\subseteq [n]=\{1,2,\dots,n\}$03, the number of representations

$S\subseteq [n]=\{1,2,\dots,n\}$04

counted up to permutation, is at most $S\subseteq [n]=\{1,2,\dots,n\}$05. When $S\subseteq [n]=\{1,2,\dots,n\}$06 and $S\subseteq [n]=\{1,2,\dots,n\}$07, this is exactly the classical multiplicative Sidon condition (Jing et al., 2022). Among the results in this direction, it is proved that there exist absolute constants $S\subseteq [n]=\{1,2,\dots,n\}$08 and $S\subseteq [n]=\{1,2,\dots,n\}$09 such that for every $S\subseteq [n]=\{1,2,\dots,n\}$10 and every finite $S\subseteq [n]=\{1,2,\dots,n\}$11, one can find a multiplicative $S\subseteq [n]=\{1,2,\dots,n\}$12-subset of size

$S\subseteq [n]=\{1,2,\dots,n\}$13

with the specific refinement that for $S\subseteq [n]=\{1,2,\dots,n\}$14 one may take $S\subseteq [n]=\{1,2,\dots,n\}$15, and for sufficiently large $S\subseteq [n]=\{1,2,\dots,n\}$16 one may take $S\subseteq [n]=\{1,2,\dots,n\}$17 (Jing et al., 2022).

There is also a fixed-multiplier notion. A set $S\subseteq [n]=\{1,2,\dots,n\}$18 is $S\subseteq [n]=\{1,2,\dots,n\}$19-multiplicative if $S\subseteq [n]=\{1,2,\dots,n\}$20 for all $S\subseteq [n]=\{1,2,\dots,n\}$21, and more generally $S\subseteq [n]=\{1,2,\dots,n\}$22-multiplicative if $S\subseteq [n]=\{1,2,\dots,n\}$23 implies simultaneously $S\subseteq [n]=\{1,2,\dots,n\}$24 and $S\subseteq [n]=\{1,2,\dots,n\}$25 for all $S\subseteq [n]=\{1,2,\dots,n\}$26, $S\subseteq [n]=\{1,2,\dots,n\}$27, and $S\subseteq [n]=\{1,2,\dots,n\}$28 (Wakeham et al., 2011). Wakeham and Wood determined the maximum density of an $S\subseteq [n]=\{1,2,\dots,n\}$29-multiplicative set in $S\subseteq [n]=\{1,2,\dots,n\}$30: if $S\subseteq [n]=\{1,2,\dots,n\}$31, then the maximum density is

$S\subseteq [n]=\{1,2,\dots,n\}$32

This is qualitatively different from the classical multiplicative Sidon problem, where the best-known constructions have zero asymptotic density (Wakeham et al., 2011). A common confusion is to treat these fixed-multiplier avoidance problems as equivalent to the classical pairwise-product uniqueness problem; they are not.

6. Extraction, obstruction, and broader combinatorial context

A separate line of work asks not for extremal subsets of $S\subseteq [n]=\{1,2,\dots,n\}$33, but for large multiplicative Sidon-type subsets inside an arbitrary finite set $S\subseteq [n]=\{1,2,\dots,n\}$34. Shkredov proved a structural dichotomy based on higher energies: for any finite $S\subseteq [n]=\{1,2,\dots,n\}$35 or $S\subseteq [n]=\{1,2,\dots,n\}$36 with $S\subseteq [n]=\{1,2,\dots,n\}$37 and any $S\subseteq [n]=\{1,2,\dots,n\}$38, there exists $S\subseteq [n]=\{1,2,\dots,n\}$39 such that either $S\subseteq [n]=\{1,2,\dots,n\}$40 has a highly structured large subset with small doubling, or $S\subseteq [n]=\{1,2,\dots,n\}$41 contains either an additive Sidon-type subset or a multiplicative Sidon-type subset of size $S\subseteq [n]=\{1,2,\dots,n\}$42 for some $S\subseteq [n]=\{1,2,\dots,n\}$43 (Shkredov, 2021). In the low-energy regime, the mechanism is probabilistic deletion from a random sample; in the high-energy regime, one first extracts structured sets and then uses incidence or sum–product estimates to force small multiplicative higher energies (Shkredov, 2021).

A complementary result of Balogh, Bush, Croot, Das, and Strauch shows that every finite set of integers contains a comparatively large multiplicative $S\subseteq [n]=\{1,2,\dots,n\}$44-subset. In particular, for $S\subseteq [n]=\{1,2,\dots,n\}$45, every finite $S\subseteq [n]=\{1,2,\dots,n\}$46 contains a multiplicative $S\subseteq [n]=\{1,2,\dots,n\}$47-subset $S\subseteq [n]=\{1,2,\dots,n\}$48 with

$S\subseteq [n]=\{1,2,\dots,n\}$49

and the proof of this case uses a point–hyperbola incidence theorem for solutions to

$S\subseteq [n]=\{1,2,\dots,n\}$50

(Jing et al., 2022).

These positive extraction results are balanced by sharp obstructions. Roche-Newton and Warren constructed an infinite set $S\subseteq [n]=\{1,2,\dots,n\}$51 such that any subset $S\subseteq [n]=\{1,2,\dots,n\}$52 with $S\subseteq [n]=\{1,2,\dots,n\}$53 is not a multiplicative Sidon set, thereby refuting a conjecture of Klurman and Pohoata (Roche-Newton et al., 2021). Their finite model is

$S\subseteq [n]=\{1,2,\dots,n\}$54

where $S\subseteq [n]=\{1,2,\dots,n\}$55 is the set of primes $S\subseteq [n]=\{1,2,\dots,n\}$56 and $S\subseteq [n]=\{1,2,\dots,n\}$57 is the set of primes $S\subseteq [n]=\{1,2,\dots,n\}$58. A subset $S\subseteq [n]=\{1,2,\dots,n\}$59 is multiplicative Sidon exactly when the corresponding bipartite graph on $S\subseteq [n]=\{1,2,\dots,n\}$60 is $S\subseteq [n]=\{1,2,\dots,n\}$61-free, because a non-trivial equality

$S\subseteq [n]=\{1,2,\dots,n\}$62

is precisely a $S\subseteq [n]=\{1,2,\dots,n\}$63-cycle. The Kővári–Sós–Turán theorem then bounds the size of multiplicative Sidon subsets of $S\subseteq [n]=\{1,2,\dots,n\}$64 by the $S\subseteq [n]=\{1,2,\dots,n\}$65-power barrier (Roche-Newton et al., 2021).

Taken together, these results place multiplicative Sidon sets at the center of a broader extremal theory. On one side are constructions and extraction theorems driven by energy decompositions, incidence estimates, and graph sparsity; on the other are obstruction examples showing that very large multiplicative Sidon substructures need not exist inside arbitrary ambient sets. The most recent progress on gaps adds analytic prime-distribution phenomena to this picture, indicating that the next advances are likely to come from tighter interaction between multiplicative combinatorics and short-interval prime theory (Doorn et al., 5 Jun 2026).