Dyson's transform

Freeman Dyson in 2005

Dyson's transform is a fundamental technique in additive number theory. It was developed by Freeman Dyson as part of his proof of Mann's theorem,Template:Rp 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.Template:Rp The term Dyson's transform for this technique is used by Ramaré.Template:Rp Halberstam and Roth call it the τ-transformation.Template:Rp

This formulation of the transform is from Ramaré.Template:Rp 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 $A^{\prime }=A\cup (B+\{e\})$ and $B^{\prime }=B\cap (A-\{e\})$. The transformed sequences have the properties:

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

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.

References

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