tag:blogger.com,1999:blog-3722233.post4036509745716529498..comments2019-12-12T06:26:43.248-05:00Comments on Computational Complexity: Two cutoffs about Warings problem for cubesLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3722233.post-66400362682054306562015-04-16T08:41:01.982-04:002015-04-16T08:41:01.982-04:00Let A be a random subset of integers with the same...Let A be a random subset of integers with the same density as cubes: each natural n is included in A with probability n^(-2/3). Let Y(x) be a random variable indicating now many integers less than x cannot be presented as a sum of 7 elements of A. You statement "all but y numbers ≤ x are the sum of 7 cubes" becomes "interesting" if y is substantially different from EY(x). To make the word "interesting" more concrete, we can set up a limit in say 0.01, that is, the statement is interesting if and only if P(|Y(x)-EY(x)| > |y-EY(X)|) > 0.01. This indicate that with 99% chance there is some "interesting" structure distinguising additive properties of cubes from random sets with the same density. Once this structure is identified, it can be included into the definition of A (now it is a random set subject to this structure) and the process can be repeated. Bogdannoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-45355041861119030062015-04-12T09:17:15.932-04:002015-04-12T09:17:15.932-04:00If we're allowing cubes of negative numbers, t...If we're allowing cubes of negative numbers, then 23 is representable as well.<br />23 = 27 - 8 + 1 + 1 + 1 + 1 (+ 0 + 0)Andy Parrishhttps://www.blogger.com/profile/12252029594014518238noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-36793819541690364862015-04-11T14:48:20.344-04:002015-04-11T14:48:20.344-04:00I think how interesting the data is depends on the...I think how interesting the data is depends on the data, not the size of the data. If the largest exception is N and you have results up to 2N, it's hard to say you have evidence that there are no further exceptions greater than 2N. But if the largest exception is N and you have results up to 1000N, that's interesting.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-50189756634369751512015-04-08T06:41:12.568-04:002015-04-08T06:41:12.568-04:00239 cannot be represented by the sum of 8 natural ...239 cannot be represented by the sum of 8 natural cubes, but can be represented by 8 integer cubes.<br />239 = 512 - 343 + 125 - 81 + 27 - 1Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-3722233.post-68907589951959958232015-04-07T13:55:23.125-04:002015-04-07T13:55:23.125-04:00All integers except 23 and 239 are sums of eight c...All integers except 23 and 239 are sums of eight cubes. https://projecteuclid.org/euclid.bams/1183502007Anonymousnoreply@blogger.com