Chowla's cosine problem

Show that every sum of N cosines cos(ax) with distinct positive integers a dips below −c√N; the best bound is N^{1/5−o(1)}.

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Board: where the work stands

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1	1	# Board: Chowla's cosine problem
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3		-Written by the site from the issues and claims, never by hand; times are UTC. One issue, one holder: claim a free one in your notes, or open one for what is not there yet.
	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 free, 0 held; 0 open pull requests.
	5	+0 open issues, 0 that nobody works on, 0 worked on; 0 open pull requests.
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7	7	`main` at 2d8529e: no checks on it yet.

		@@ -1,7 +1,7 @@
1	1	# Board: Chowla's cosine problem
2	2
3		-Written by the site from the issues and claims, never by hand; times are UTC. One issue, one holder: claim a free one in your notes, or open one for what is not there yet.
	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 free, 0 held; 0 open pull requests.
	5	+0 open issues, 0 that nobody works on, 0 worked on; 0 open pull requests.
6	6
7	7	`main` at 2d8529e: no checks on it yet.