Dyson's transform

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

Freeman Dyson in 2005

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 $A=\{a_{1}<a_{2}<\cdots \}$ and $B=\{0=b_{1}<b_{2}<\cdots \}$ 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 $A'=A\cup (B+\{e\})$ and $\,B'=B\cap (A-\{e\})$ . The transformed sequences have the properties:

$A'+B'\subset A+B$
$\{e\}+B'\subset A'$
$0\in B'$
$A'(m)+B'(m-e)=A(m)+B(m-e)$

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

$A_{1}+B_{1}\subset A+B,A_{2}+B_{2}\subset A+B$
$|A_{1}|+|A_{2}|=2|A|$ , $|B_{1}|+|B_{2}|=2|B|$

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

References

Additive Number Theory: Inverse Problems and the Geometry of Sumsets By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, ISBN 0-387-94655-1, https://books.google.com/books?id=PqlQjNhjkKUC&q=%22e-transform%22 p. 42
Halberstam, H.; Roth, K. F. (1983). Sequences (revised ed.). Berlin: Springer-Verlag. ISBN 978-0-387-90801-4.
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.
DeVos, Matt (2016). "On a Generalization of the Cauchy-Davenport Theorem". Integers. 16.

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[nathanson-1] Additive Number Theory: Inverse Problems and the Geometry of Sumsets By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, ISBN 0-387-94655-1, https://books.google.com/books?id=PqlQjNhjkKUC&q=%22e-transform%22 p. 42

[Sequences-2] Halberstam, H.; Roth, K. F. (1983). Sequences (revised ed.). Berlin: Springer-Verlag. ISBN 978-0-387-90801-4.

[Ramaré-3] 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.