MultiplicativeEnergy

The number of quadruples $(a_1,a_2,b_1,b_2)\in A\times A\times B\times B$ with $a_1b_1=a_2b_2$ .

mathlib4Finset.mulEnergy

MultiplicativeEnergy(A, B)

E^\times(A,B)=\#\{(a_1,a_2,b_1,b_2): a_1b_1=a_2b_2\}

, the pairs of pairs with equal products.

MultiplicativeEnergy(A)

E^\times(A,A)

.

Typed signatures

Type	Package	Applies
`(collection, collection?) -> integer`	`combinatorics`	always

Domain: Collections

Details

Defined in Epsil as $\sum_x r(x)^2$ , where $r(x)$ is the number of pairs $(a,b)\in A\times B$ with $ab=x$ : AdditiveEnergy with products for sums.
Zero is an ordinary member. It multiplies every member to $0$ , so all of those pairs share the one product, which contributes $r(0)^2$ : $\{0,1,2,3\}$ has $r(0)=7$ .
The members are the distinct members of the collection, as in mathlib's Finset: a list's repeats are ignored, and a Set reads like a list.
Members must be exact rationals. A float, a symbol or an infinite collection leaves the call unevaluated.
A geometric progression has the energy of an arithmetic one of the same length: $\{1,2,4,8,16\}$ and $\{0,\dots,4\}$ both give $85$ . The primes are a multiplicative Sidon set, by unique factorization.
Symmetric in its two arguments, and unchanged by scaling either set by a nonzero rational.
Not a Wolfram, SymPy or Sage head.

Examples

1 more kept as data, checked against the oracles

Basic #

the quadruples of $\{1,\dots,5\}$ with $a_1b_1=a_2b_2$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack1, 2, 3, 4, 5\bigr\rbrack)

Out

49

powers of 3 multiply as their exponents add: $\{0,1,2\}$ and $\{0,1\}$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack1, 3, 9\bigr\rbrack, \bigl\lbrack1, 3\bigr\rbrack)

Out

10

Scope #

one member gives one quadruple, zero included

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack0\bigr\rbrack)

Out

1

no members, no quadruples

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack\bigr\rbrack)

Out

0

exact rationals count too: $\{2^{-1},2^0,2^1\}$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack\frac{1}{2}, 1, 2\bigr\rbrack)

Out

19

Properties #

zero multiplies every member to $0$ : those $7$ pairs agree with each other, $7^2$ quadruples, and the other nine pairs add $15$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack0, 1, 2, 3\bigr\rbrack)

Out

64

a geometric progression has the additive energy of $\{0,\dots,4\}$ , $(2\cdot 5^3+5)/3$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack1, 2, 4, 8, 16\bigr\rbrack)

Out

85

unique factorization makes the primes a multiplicative Sidon set: $2\cdot 4^2-4$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack2, 3, 5, 7\bigr\rbrack)

Out

28

a subgroup attains the maximum $|A|^3$

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack-1, 1\bigr\rbrack)

Out

8

swapping the two sets changes nothing

In

\mathrm{MultiplicativeEnergy}(\bigl\lbrack1, 3\bigr\rbrack, \bigl\lbrack1, 3, 9\bigr\rbrack)

Out

10

About

Examples	11
Checked against	mathlib4 11 agree
Crosswalk	mathlib4

Implementation

mappedmathlib4packages/oracle/src/mappings.ts

mathlib's energy is over Finsets, so each list is read as a Finset of rationals (duplicates collapse, as ours do)

mappedmathlib4packages/oracle/src/mappings.ts

mathlib's energy is over Finsets, so each list is read as a Finset of rationals (duplicates collapse, as ours do)

See also: AdditiveEnergy, Tally, Unique