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}
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 and 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 and . The transformed sequences have the properties:
References
- ^ ^{a} ^{b} 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
- ^ ^{a} ^{b} Halberstam, H.; Roth, K. F. (1983). Sequences (revised ed.). Berlin: Springer-Verlag. ISBN 978-0-387-90801-4.
- ^ ^{a} ^{b} ^{c} 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.
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