Dyson's transform

{{Short description|Mathematical tool}}

File:Freeman Dyson.jpg

Dyson's transform is a fundamental technique in additive number theory.{{Cite book |last=Nathanson |first=Melvyn B. |url=https://books.google.com/books?id=PqlQjNhjkKUC&q=%22e-transform%22 |title=Additive Number Theory: Inverse Problems and the Geometry of Sumsets |date=1996-08-22 |publisher=Springer Science & Business Media |isbn=978-0-387-94655-9 |language=en}} It was developed by Freeman Dyson as part of his proof of Mann's theorem,{{cite book | last1 = Halberstam | first1 = H. |authorlink1 = Heini Halberstam | last2 = Roth | first2 = K. F. | authorlink2 =Klaus Roth | title = Sequences | title-link=Sequences (book) | publisher = Springer-Verlag | location = Berlin | year = 1983 | edition = revised | isbn = 978-0-387-90801-4 }}{{Rp|17}} is used to prove such fundamental results of additive number theory as the Cauchy-Davenport theorem, 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.{{cite journal | author=O. Ramaré | authorlink=Olivier Ramaré | title=On šnirel'man's constant | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | volume=22 | year=1995 | issue=4 | pages=645–706 | url = http://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0 | accessdate = 2009-03-13}}{{Rp|700–701}} The term Dyson's transform for this technique is used by Ramaré.{{Rp|700–701}} Halberstam and Roth call it the τ-transformation.{{Rp|58}}

This formulation of the transform is from Ramaré.{{Rp|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 + 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 $A\text{'}=\; A\; \backslash cup\; (B\; +\; \backslash \{e\backslash \})$ and $\backslash ,B\text{'}=\; B\; \backslash cap\; (A\; -\; \backslash \{e\backslash \})$. The transformed sequences have the properties:

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

Other closely related transforms are sometimes referred to as Dyson transforms. This includes the transform defined by $A\_1\; =\; A\; \backslash cap\; (A\; +\; \backslash \{e\backslash \})$, $A\_2\; =\; A\; \backslash cup\; (A\; +\; \backslash \{e\backslash \})$, $B\_1\; =\; B\; \backslash cap\; (-\backslash \{e\backslash \}\; +\; B)$, $B\_2\; =\; B\; \backslash cup\; (-\backslash \{e\backslash \}\; +\; B)$ for $A,\; B$ sets in a (not necessarily abelian) group. This transformation has the property that

• $A\_1\; +\; B\_1\; \backslash subset\; A\; +\; B,\; A\_2\; +\; B\_2\; \backslash subset\; A\; +\; B$
• $|A\_1|\; +\; |A\_2|\; =\; 2|A|$, $|B\_1|\; +\; |B\_2|\; =\; 2|B|$

It can be used to prove a generalisation of the Cauchy-Davenport theorem.{{Cite journal |last=DeVos |first=Matt |date=2016 |title=On a Generalization of the Cauchy-Davenport Theorem |url=http://math.colgate.edu/~integers/q7/q7.Abstract.html |journal=Integers |volume=16}}