Incidentally Salem's book on "Algebraic Numbers and Fourier Analysis" (despite its title) includes a wealth material of general interest which --- as far as I know --- is found nowhere else: for example, a neat elementary proof using only GCD's that a linear recurring sequence over the integers has finitely many zeros not lying in regularly-spaced zero subsequences. WFL On 9/12/13, Bill Gosper <billgosper@gmail.com> wrote:
PVS numbers? http://en.wikipedia.org/wiki/Pisot–Vijayaraghavan_number http://en.wikipedia.org/wiki/Salem_number WFL On 9/11/13, Bill Gosper < billgosper@gmail.com> wrote:
DanA> When I try to evaluate that exact sum in Mma, it gives me only numbers that are to > 20 decimal places equal to 0. (Adding only 1000 or 10000 terms, asking for 10 digits' precision.) I'd guess only a set of measure 0 of real numbers x would have their Sum[Round[x^n]-x^n,{n,0,∞}] <> 0. But it's easy to believe that many algebraic numbers don't. What's your secret? --Dan
On 2013-09-10, at 8:11 PM, Bill Gosper wrote: Let t:=1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)), the tribonacci constant. Then, empirically, Sum[Round[x^n]-x^n,{n,0,∞}] Gaa, I meant Round[t^n]-t^n (Mac OS crashed as I prepared to send.) ==-0.80851211604688125... -2 + ((3 Sqrt[3] - Sqrt[11])^(1/3) + (3 Sqrt[3] + Sqrt[11])^(1/3))/(2^(2/3) Sqrt[3])
In[2]:= NSum[Round[GoldenRatio^k] - GoldenRatio^k, {k, 288}, NSumTerms -> 288, WorkingPrecision -> 99]
Out[2]= 0.618033988749894848204586834365638117720309179806089199107257598931385927496544945583310480117151672 In[3]:= (% + 1/2)^2 Out[3]= 1.2500000000000000000000000000000000000000000000007297116525363033564609 5964404522000200488900285933 This can't be new. Anybody? --rwg _________________________________ Thanks, Fred! So, defining roundoff[x]:=FractionalPart[x+1/2]-1/2 (or Round[x]-x), why has nobody thought to Sum[roundoff[Pisot^k]]? At first I thought it's because of the modern habit of norming everything. But Sum[Abs[roundoff[Pisot^k]]] is quadratic when Pisot is, so somebody should have noticed.
This is and endless source of identities. It seems that, if it converges, Sum[roundoff[Pisot1^k]^p1*roundoff[Pisot2^k]^p2.../r^k] is algebraic with the same degree as Pisot1 Pisot2 ... . E.g., NSum[(FractionalPart[(1/3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^k + 1/2] - 1/2)^2/2^k, {k, 288}, NSumTerms -> 288, WorkingPrecision -> 288];
In[31]:= RootApproximant[%]
Out[31]= Root[-215317 + 2949880 #1 + 1899072 #1^2 + 326144 #1^3 &, 1]
In[29]:= RootApproximant[NSum[(FractionalPart[(1/ 3 (1 + (19 - 3 Sqrt[33])^(1/3) + (19 + 3 Sqrt[33])^(1/3)))^ k + 1/2] - 1/2)*(FractionalPart[GoldenRatio^k + 1/2] - 1/2), {k, 288}, NSumTerms -> 288, WorkingPrecision -> 105]]
Out[29]= Root[ 41 + 567 #1 - 144 #1^2 - 313 #1^3 - 8 #1^4 + 55 #1^5 + 11 #1^6 &, 2] --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun