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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1	{
2	"domain": "mathematics",
3	"topics": [
4	"number-theory",
5	"combinatorics"
6	],
7	"type": "construction",
8	"also": [
9	"proof"
10	],
11	"source": {
12	"key": "erdos",
13	"name": "Erdős Problems",
14	"url": "https://www.erdosproblems.com/39",
15	"id": "39",
16	"curator": "Thomas Bloom",
17	"poser": "Paul Erdős",
18	"status": "open, $500 prize",
19	"as_of": "2026-10-20",
20	"prize": "$500",
21	"catalog": "0521c19bbd38"
22	},
23	"review": true,
24	"approval": "first-time"
25	}