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If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available here

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here is in error in claiming O(((log log N)^5 N/log N) and that it should be O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.

If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available here

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here is in error in claiming O(((log log N)^5 N/log N) and that it should be O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.

If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available at http://arxiv.org/pdf/1405.5800v2.pdf

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here: http://arxiv.org/pdf/1405.5800v2.pdf is in error in claiming O(((log log N)^5 N/log N) and that it should be O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.

If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available here

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here is in error in claiming O(((log log N)^5 N/log N) and that it should be O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.

If a sequence avoids three term arithmetic progressions it is less than ((log log n)^4)N/log N according to "A Quantitative improvement for Roths’S Theorem On Arithmetic Progressions" which is available at http://arxiv.org/pdf/1405.5800v2.pdf

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here: http://arxiv.org/pdf/1405.5800v2.pdf is in error in claiming O(((log log n)^5 N/log N) and that it should be O(((log log n)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log n to the fourth power better than N/log N if the preprint in the first paragraph is correct.

If a sequence avoids three term arithmetic progressions it is less than ((log log N)^4)N/log N according to "A Quantitative improvement for Roths’s Theorem On Arithmetic Progressions" which is available at http://arxiv.org/pdf/1405.5800v2.pdf

The above preprint also claims that the result of "On Roth’s Theorem On Progressions" availabe on arxiv here: http://arxiv.org/pdf/1405.5800v2.pdf is in error in claiming O(((log log N)^5 N/log N) and that it should be O(((log log N)^6 N/log N).

So the upper bound even for three term arithmetic progressions is at most a factor of log log N to the fourth power better than N/log N if the preprint in the first paragraph is correct.