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

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

References

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&dq=%22e-transform%22&source=gbs_navlinks_s, p. 42
2. Halberstam, H.; Roth, K. F. (1983). Sequences (revised ed.). Berlin: Springer-Verlag. ISBN 978-0-387-90801-4.
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.