5 M ay 2 00 8 Square - Difference - Free Sets of Size Ω ( n 0 . 7334 ) Richard Beigel
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چکیده
A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x 6= y, such that |x − y| is a square. Let sdf(n) be the size of the largest SDF subset of {1, . . . , n}. It is known that n 0.733077... ≤ sdf(n) ≤ O (
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8 M ay 2 00 8 Square - Difference - Free Sets of Size Ω ( n 0 . 7334
A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x 6= y, such that |x − y| is a square. Let sdf(n) be the size of the largest SDF subset of {1, . . . , n}. Ruzsa has shown that sdf(n) = Ω(n65 ) = Ω(n0.733077···) We improve on the lower bound by showing sdf(n) = Ω(n205 ) = Ω(n0.7334···) As a corollary we obtain a new lower bound on the quadratic van der Waer...
متن کاملA pr 2 00 8 Square - Difference - Free Sets of Size Ω ( n 0 . 7167 − ǫ )
A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x 6= y, such that |x − y| is a square. Let sdf(n) be the size of the largest SDF subset of {1, . . . , n}. It is known that n ≤ sdf(n) ≤ O ( n(log log n)2/3 (log n)1/3 )
متن کاملSquare-Difference-Free Sets of Size Ω(n0.7334···)
A set A ⊆ N is square-difference free (henceforth SDF) if there do not exist x, y ∈ A, x 6= y, such that |x− y| is a square. Let sdf(n) be the size of the largest SDF subset of {1, . . . , n}. Ruzsa [10] has shown that proved sdf(n) ≥ Ω(nlog65 7) ≥ Ω(n0.733077···). sdf(n) = Ω(n65 ) = Ω(n0.733077···) We improve on the lower bound by showing sdf(n) = Ω(n205 ) = Ω(n0.7334···) As a corollary we obt...
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