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:

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

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

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

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