Consiering bouns for approximation of to (version. Abstract: Estimating bouns of best approximations of to is iscusse. In the first part I evelop a powerseries, which shoul give practicable limits for that bouns in terms of an. In the secon part the relation to the -cycle-question in the Collatz-problem is referre, as well as a connection to an unsolve aspect of the Waring-problem. he article is a still unfinishe raft; it is an -a bit upate version- of a manuscript of 00, publishe in sci.math an e.sci.mathemati-newsgroups. Gottfrie Helms helms(atuni-assel.e.0.00. Expression as powerseries o etermine the bouns for best approximations of to in terms of (. / --->, where also is a function of, I originally state the problem in this way: (. // terms where the number of 's an of 's in the numerator are chosen, such that the partial proucts are best approximate to (. eps > 0 in the sense that eps>0 is minimum. he logarithm is then (. log log where, Expresse in the series-representation for logarithms this is: (. Rearranging into two partial series accoring to the signs: (.
Consiering bouns for approximation of to. -- Aapting the terms to match the even powers of / an to get lie enominators: (. gives rearrange an finally encoe as a general term: (. > 0 ( ( ow, to rewrite this in terms of an, with (. / / / we have that (0. an an thus (. -, - the above gives (. > ( ( 0 or, setting -/, (. ( ( Given the last formula, the problem is now a question of approximating a powerseries, with an appropriate application of analysis of the ratio of the remainer an the current partial approximation. Written in a base--igitsystem, where unfortunately the leaing igits are unnormalize, one can iscuss, at which term the absolute value of the igits are normalize an cannot, for instance prouce carries to the left. For instance, with, this gives, with the fractions as (unnormalize base--igits:
Consiering bouns for approximation of to. -- (. 0. ( base 0 an it is immeiately visible, that the approximation cannnot be better than with the first igit alone, since the sequence of partial sums increases. o get approximations near to zero we nee, that the numerators are positive in the leaing igits an then turn to become negative in the trailing igits, but are still large enough to compensate for the alreay achieve partial sum. his can be configure by increasing relatively to. (But note, that the enominators of each term are inepenent of, thus constituting a minimal "granularity" of summans. For instance we use,. hen we have -- an the igit-representation loos lie (. 0. ( base 0 0. ( base 0 an the partial sums converge quicly to /, /-/², where the anylysis of the remainer shoul show, that we cannot be better than something lie / or /0 alreay after the secon step,. he approximation of the weighte logarithm series recompute to the original problem nees then exponentiation of the so-foun epsilon an may exhibit the reason for the ba approximability. I couln't finish the estimation of the epsilon for the logarithm-series yet, so also a efinitive estimation for the original question is still unanswere. I'm currently unable to consier this in more etail, but it may be an entrypoint for a more experience reaer.. Connection to the Collatz--cycle an Waring-problem he original question of this approximation arose in the stuy of -cycles ("primitive loop" in my woring, an whether the next perfect power of greater than ("powerceil( " can be smaller than the rhs in the following equation (. powerceil ( < ( which, when (in the sense of maximal probability of a solution is set to, reuces to (. powerceil( < ( ( ( or in the igit-system base, with [ / ] (where the bracets enote the integral part an r the remainer, so that an (. r r. 000. ( base A A (. powerceil ( 0000. 0. hen (. becomes (0. powerceil( < (.. ( base an lhs an rhs must even be equal, without trailing igits, such that
Consiering bouns for approximation of to. -- (. A 0. 0000... r.rrrrrrr... ( base his can only become true, if the sum r- - prouces not only a carry, but where also this carry zeroes each igit an A -, thus if for at least one it must occur, that exactly r, A - an r - A to mae a Collatz--cycle possible. o show, that this cannot happen means also to complete the Waring-conjecture, accoringly to http:/mathworl.wolfram.com in the page about "powerfractions". his property is surprisingly something special as can be seen in a comparison of graphs. Here I plotte against r for ifferent numbers. Here [/ ], r (mo, an the re points are plotte for to, where xr an y For the sum to prouce a carry means, that a point occurs above the main iagonal. o point is in the upper triangle;except the special point for in the mile. An arbitrary number in the near of inserte in the above formula replacing. All combinations of an r seem to occur. Here times the "golen ratio" (phi is use (~., which is also in the near of. here is exceptional clarity, thus I assume, for phi this type of graph shows another special approximation behaviour. (ore pictures of that can be seen in: http://go.helms-net.e/math/collatz/loopintro/graphson_approximations.htm main page: http://go.helms-net.e/math/collatz/loopintro/collloopintro_main.htm
Consiering bouns for approximation of to. --. Conclusion he iscussion of the approximation as one in the first part of this article is relate to this only as a byprouct. While the argument here in the secon part focuses, that an approximation cannot approach or even be better than / (which woul be neee to satisfy the rhs of eq(. an thus allow a -cycle in the Collatz-problem, I trie to setup the arguments in the first part of this article to explain, that (an why the bouns of best approximations are even much worse. I assume that it can be boune by some low-orer-polynomial in, lie / eps (where eps<<. or a logarithmic term lie /( ln( using better nowlege about an appropriate techniques for approximations of powerseries. Gottfrie Helms
Consiering bouns for approximation of to. --. Appenix: able of unnormalize igits unnormalize igits of / powerceil( / (base unnormalize "igits" at position (beyon ecimal point, Goo approximations can only be achieve, if o leaing igits are positive, trailing igits are negative o the change of sign occurs in an early region (small, so that the remaining -strongly ecreasing- tail of the series can still compensate the accumulate leaing partial sum o an still the no-carry- point is as far as possible from the ecimal point Goo approximations occur at, (last significant term, very goo approximations occur at (last significant term an generally relate to high values in the continue-fractions-expansion. Pari/GP-coe: b /; n; ^ceil(bn/^n \\ 0.0000000 n;^ceil(bn/^n \\ 0.00000 n;^ceil(bn/^n \\ 0.0000 lm,n local(su,s,t,; \ sm-n;tn-m; m-n; \ su sum(,00,(n/((-/(^; \ return (su.0; n ;lceil(nb,n \\ 0.0000000 n;lceil(nb,n \\ 0.00000 n;lceil(nb,n \\ 0.0000