Timeline for How small can a sum of a few roots of unity be?
Current License: CC BY-SA 3.0
12 events
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| Jan 9, 2017 at 17:12 | comment | added | kodlu | It seems that the least number of distinct $n$th roots of unity summing to the smallest possible nonzero magnitude is growing with something like linear growth with $n$. This looks like an interesting question in itself. | |
| Oct 16, 2015 at 18:55 | comment | added | Todd Trimble | @GerryMyerson Just wish to say thank you very much for the additional information! | |
| Oct 9, 2015 at 2:02 | history | edited | Gerry Myerson | CC BY-SA 3.0 |
summarized contents of paper
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| Oct 9, 2015 at 0:58 | comment | added | Todd Trimble | I hate to bother @GerryMyerson with this, but since the author noobcake hasn't been here in almost 5 years, I don't know who else to ask. The link is not open-access; would it be possible to state briefly the relevant result from that paper in the answer here? | |
| Nov 16, 2010 at 18:35 | comment | added | tdnoe | The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude. There is also a plot of the least magnitude for n up to 81. | |
| Nov 15, 2010 at 5:05 | vote | accept | Terry Tao | ||
| Nov 15, 2010 at 4:46 | comment | added | Aaron Meyerowitz | for $C=210(2\pi)^5$ by my calculations. | |
| Nov 15, 2010 at 3:35 | comment | added | Gerry Myerson | On re-reading my paper, I find that in fact I was aware of the Graham-Sloane paper at the time, as it's in the references. Also I should note that much of the paper is about the general case, not just the case of 5 roots. In particular, for 10 roots, the equation $1^r+5^r+9^r+17^r+18^r=2^r+3^r+11^r+15^r+19^r$, which holds for $0\le r\le4$, leads, by the method explained in the paper, to a non-zero sum of 10 $n$th roots of modulus $Cn^{-5}+O(n^{-6})$ for even $n$. | |
| Nov 14, 2010 at 23:39 | comment | added | Gerry Myerson | I D Shkredov, Fourier analysis in combinatorial number theory, Russian Math Surveys 65:3 (2010) 513-567, writes on pp 544-545, "We conclude this section by discussing the question of how small the Fourier coefficient of a characteristic function can be. This problem was first considered in [132]." That's my paper, though I certainly did not use those terms. They give an estimate and source it to V F Lev, Linear equations over ${\bf F}_p$ and moments of exponential sums, Duke Math J 107 (2001) 239-263. | |
| Nov 14, 2010 at 23:34 | comment | added | Gerry Myerson | Regret to say the paper contains all I know about the question. There's a small contribution by Dean Hickerson mentioned on page 967 of Richard Guy's "Monthly Unsolved Problems, 1969-1987," Monthly 94 (Dec. 1987). An earlier reference of which I was unaware in 1986 is Graham and Sloane, Anti-Hadamard matrices, freely available at neilsloane.com/doc/1218anti.ps On p. 11, they ask, "What is the smallest magnitude of any nonvanishing sum of distinct $n$-th roots of unity?" More references next comment: | |
| Nov 14, 2010 at 21:35 | comment | added | JSE | The author of the above paper is a frequent MO commmenter, so I hope we'll have an authoritative answer shortly. Gerry Myerson, I summon thee! | |
| Nov 14, 2010 at 21:03 | history | answered | noobcake | CC BY-SA 2.5 |