Dyson's Transform

Dyson's transform is a fundamental technique in additive number theory.Additive Number Theory: Inverse Problems and the Geometry of Sumsets By Melvyn Bernard Nathanson, Springer, Aug 22, 1996, , https://books.google.com/books?id=PqlQjNhjkKUC&dq=%22e-transform%22&source=gbs_navlinks_s, p. 42 It was developed by Freeman Dyson as part of his proof of Mann's theorem, 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. The term ''Dyson's transform'' for this technique is used by Ramaré. Halberstam and Roth call it the τ-transformation.

This formulation of the transform is from Ramaré. 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 , ''x'' Suppose $A= \$ and $B= \$ 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 \$ and $\,B'= B \cap \$. The transformed sequences have the properties:

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

References

Sumsets
Freeman Dyson

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