Timeline for Is there a "complete" Sidon sequence?
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| Nov 17, 2015 at 16:25 | comment | added | Mark Lewko | A precise formulation of the result Ben references is as follows: if the representation function of a set $A$ satisfies $r(n) \leq 1$ for all $n$ then $|A \cap [1,...,n]| \leq C \sqrt{n/ \log n}$ holds infinitely often. It is open if $r(n) \leq 2$ implies $|A \cap [1,...,n]| \leq o( \sqrt{n} ) $. The problem of showing $r(n) \leq k$ implies $|A \cap [1,...,n]| \leq o( \sqrt{n} ) $ for all $k$ implies a famous longstanding problem of Erdos and Turan. See: mathoverflow.net/questions/43995/… | |
| Nov 17, 2015 at 14:00 | comment | added | Ben Green | No, there is a well-known result of Erdos (mentioned on the Wikipedia page for Sidon Sequences) saying that an infinite Sidon sequence must have liminf a_n/n^2 = 0, whereas a sequence with the property you mention cannot have this property. I believe you can find the proof in Halberstam and Richert's sequences book, though I do not currently have it to hand. | |
| Nov 17, 2015 at 12:33 | comment | added | Gerry Myerson | I think you're asking for an asymptotic basis for the integers of order 2, such that no number has more than one representation. Erdos conjectured that on the contrary in any such basis the number of representations is unbounded. I don't have Guy, Unsolved Problems In Number Theory handy, but if I did, that's the first place I'd look. In the meantime, you'll want to look at en.wikipedia.org/wiki/Erdős–Turán_conjecture_on_additive_bases | |
| Nov 17, 2015 at 10:14 | history | asked | Konstantinos Gaitanas | CC BY-SA 3.0 |