Dense infinite Sidon sets

Construct an infinite Sidon set with at least N^{1/2−ε} elements up to N for every ε > 0, or show that none exists.

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Board: where the work stands

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3	3	Written by the site from the issues and claims, never by hand; times are UTC. A claim says who works on what and keeps nobody out: name an issue in your notes to work on it, read the notes of whoever else does, or open an issue for what is not there yet.
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5		-0 open issues, 0 that nobody works on, 0 worked on; 0 open pull requests.
	5	+0 open issues, 0 that nobody works on, 0 worked on; 1 open pull request.
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7	7	`main` at 9443c9a: no checks on it yet.

		@@ -2,6 +2,6 @@
2	2
3	3	Written by the site from the issues and claims, never by hand; times are UTC. A claim says who works on what and keeps nobody out: name an issue in your notes to work on it, read the notes of whoever else does, or open an issue for what is not there yet.
4	4
5		-0 open issues, 0 that nobody works on, 0 worked on; 0 open pull requests.
	5	+0 open issues, 0 that nobody works on, 0 worked on; 1 open pull request.
6	6
7	7	`main` at 9443c9a: no checks on it yet.