Dyson's transform

Freeman Dyson in 2005

Dyson's transform is a fundamental technique in additive number theory.[1] It was developed by Freeman Dyson as part of his proof of Mann's theorem,[2]: 17  is used to prove such fundamental results of additive number theory as the Cauchy-Davenport theorem,[1] and was used by Olivier Ramaré in his work on the Goldbach conjecture that proved that every even integer is the sum of at most 6 primes.[3]: 700–701  The term Dyson's transform for this technique is used by Ramaré.[3]: 700–701  Halberstam and Roth call it the τ-transformation.[2]: 58 

This formulation of the transform is from Ramaré.[3]: 700–701  Let A be a sequence of natural numbers, and x be any real number. Write A(x) for the number of elements of A which lie in [1, x]. Suppose A = { a 1 < a 2 < ⋯ } {\displaystyle A=\{a_{1}<a_{2}<\cdots \}} and B = { 0 = b 1 < b 2 < ⋯ } {\displaystyle B=\{0=b_{1}<b_{2}<\cdots \}} are two sequences of natural numbers. We write A + B for the sumset, that is, the set of all elements a + b where a is in A and b is in B; and similarly A − B for the set of differences a − b. For any element e in A, Dyson's transform consists in forming the sequences A ′ = A ∪ ( B + { e } ) {\displaystyle A'=A\cup (B+\{e\})} and B ′ = B ∩ ( A − { e } ) {\displaystyle \,B'=B\cap (A-\{e\})} . The transformed sequences have the properties:

  • A ′ + B ′ ⊂ A + B {\displaystyle A'+B'\subset A+B}
  • { e } + B ′ ⊂ A ′ {\displaystyle \{e\}+B'\subset A'}
  • 0 ∈ B ′ {\displaystyle 0\in B'}
  • A ′ ( m ) + B ′ ( m − e ) = A ( m ) + B ( m − e ) {\displaystyle A'(m)+B'(m-e)=A(m)+B(m-e)}


Other closely related transforms are sometimes referred to as Dyson transforms. This includes the transform defined by A 1 = A ∩ ( A + { e } ) {\displaystyle A_{1}=A\cap (A+\{e\})} , A 2 = A ∪ ( A + { e } ) {\displaystyle A_{2}=A\cup (A+\{e\})} , B 1 = B ∩ ( − { e } + B ) {\displaystyle B_{1}=B\cap (-\{e\}+B)} , B 2 = B ∪ ( − { e } + B ) {\displaystyle B_{2}=B\cup (-\{e\}+B)} for A , B {\displaystyle A,B} sets in a (not necessarily abelian) group. This transformation has the property that

  • A 1 + B 1 ⊂ A + B , A 2 + B 2 ⊂ A + B {\displaystyle A_{1}+B_{1}\subset A+B,A_{2}+B_{2}\subset A+B}
  • | A 1 | + | A 2 | = 2 | A | {\displaystyle |A_{1}|+|A_{2}|=2|A|} , | B 1 | + | B 2 | = 2 | B | {\displaystyle |B_{1}|+|B_{2}|=2|B|}

It can be used to prove a generalisation of the Cauchy-Davenport theorem.[4]

References

  1. ^ a b Nathanson, Melvyn B. (1996-08-22). Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Springer Science & Business Media. ISBN 978-0-387-94655-9.
  2. ^ a b Halberstam, H.; Roth, K. F. (1983). Sequences (revised ed.). Berlin: Springer-Verlag. ISBN 978-0-387-90801-4.
  3. ^ a b c O. Ramaré (1995). "On šnirel'man's constant". Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV. 22 (4): 645–706. Retrieved 2009-03-13.
  4. ^ DeVos, Matt (2016). "On a Generalization of the Cauchy-Davenport Theorem". Integers. 16.


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