Intrinsic Approximation on Cantor-like Sets, a Problem of Mahler

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1 Intrinsic Aroximation on Cantor-like Sets, a Problem of Mahler Ryan Broderick, Lior Fishman, Asaf Reich and Barak Weiss July 200 Abstract In 984, Kurt Mahler osed the following fundamental question: How well can irrationals in the Cantor set be aroximated by rationals in the Cantor set? Towards such a theory, we rove a Dirichlet-tye theorem for this intrinsic diohantine aroximation on Cantor-like sets. The resulting aroximation function is analogous to that for R d, but with d being the Hausdorff dimension of the set, and logarithmic deendence on the denominator instead. Keywords: Cantor sets, Diohantine Aroximation, Dirichlettye Theorem Introduction The diohantine aroximation theory of the real line is classical, extensive, and essentially comlete as far as characterizing how well real numbers can be aroximated by rationals (Schmidt 980 is a standard reference). The basic result on aroximability of all reals is Theorem (Dirichlet s Aroximation Theorem). For each x R, and for any Q N, there exist, q N, q Q, such that x < q qq. Corollary. For each x R, there exist infinitely many, q N satisfying x < q q 2. It is a classical result of Hurwitz that Corollary is false if is relaced by any constant less than 5. Furthermore the rate cannot be imroved for most irrationals, in the sense that if the exonent 2 is relaced by any real number greater than 2, the resulting set for which the inequality holds (VWA) is null. In addition, the subject of aroximating oints on fractals by rationals

2 Intrinsic Aroximation on Cantor-like Sets 2 has been extensively studied in recent years; see for examle Kleinbock-Weiss (2005), Kristensen et al. (2006), and Fishman (2009) for the BA case, and Weiss (200) and Kleinbock et al. (2004) for VWA. In 984, Mahler osed a natural roblem concerning not whether the metric roerties of these diohantine sets are reserved under intersection with fractals, but studying aroximation of irrationals in the fractal by rationals also in the fractal. As far as the authors are aware, nothing is known about such intrinsic diohantine aroximation on fractals (though there is such research for algebraic varieties, for examle Ghosh et al. 200). We resent a first ste towards an intrinsic theory determining the analogous rate of aroximation for the Cantor set C (and those constructed similarly). Namely, our result imlies Corollary 2. For all x C, there exist infinitely many solutions N, q N, q C to x < q q(log 3 q), /d where d = dim C. We conjecture that the set of numbers x satisfying, for some ɛ > 0, x < q q +ɛ (log 3 q) /d for infinitely many and q is null with resect to the standard measure on C. This would be analogous to the classical setting, where VWA has zero Lebesgue measure. (See section 3.) 2 Theorem Let C denote the Cantor-like set consisting of numbers in I = [0, ] which can be written in base b > 2 using only the digits in S {0,...b }, where S = a >. Obviously this is equivalent to artitioning I into b equal subintervals, only keeing those indexed by S, and successively continuing this on each remaining subinterval. (The usual middle-thirds set is given by b = 3, S = {0, 2}.) Throughout the aer we denote by {x} the fractional art of x and by x the integer art. To state the theorem we also associate to the set the number b 0, where b 0 is the least integer such that C is invariant under x {b 0 x} (as we shall see, this is simly equal to b excet when the given definition of C is redundant in a certain sense). Theorem 2 (Dirichlet for Cantor sets). For all x C, for every Q of the form b n, there exists /q Q C, such that x < q qq, where q Q if C = I or q b Qd 0 otherwise, where d = dimc and b = b r 0 for some r N.

3 Intrinsic Aroximation on Cantor-like Sets 3 Remark. Notice it follows from the statement that for any Q N, the same statement holds with the bound multilied by b (by letting b n Q < b n+.) Proof. First we need a characterization of the rationals in C: Lemma. A rational number is in C if and only if it can be written either as a terminating base-b exansion (left end oints in the construction) or as ( k+l i=0 c i b k+l i k j=0 c jb k j) b k+l b k. () where l, k N, and c i S. Equivalently () can also be exressed in terms of base-b exansions, i.e., ( (c0 c...c k+l ) b (c 0 c...c k ) b ) b k+l b k. (2) Proof. A rational in C has either a terminating b-ary exansion (consisting of digits from S) or an eventually eriodic one. If it is urely eriodic of eriod l, it has the form n= n= b l x 0 + b l 2 x x l (b l ) n = bl x 0 + b l 2 x x l b l, where the digits x i are in S. If the rational is not urely eriodic then one must insert some number k of initial zeros and then add the initial terminating exansion of length k, so we obtain the form b l x 0 + b l 2 x x l (b l )b k + (bl )(b k y 0 + b k 2 y y k ) (b l )b k for some digits y i S. Rearranging this gives the result. Now let x C \ Q. Given any n N, let Q = b n. Denote by M = (m i ) the semigrou of ositive integer multilication mas mod leaving C invariant. We order the elements in increasing order from = m 0. There are a n ossibilities for the first n digits in the b-ary exansion of {qx}. Consider the elements of C given by 0, {m 0 x},...{m a n x}. By the igeonhole rincile either there exist 0 q, q < a n such that {m q x} and {m q x} have the same first n digits, or the same holds for some 0 q < a n and 0. That is, they are in the same interval of the n-th stage of C s construction. Assuming the former, it follows that {m q x} {m q x} < Q. Rewriting gives mq x m q x m q x + m q x < Q. Setting = m q x m q x we get x < m q m q (m q m q )Q.

4 Intrinsic Aroximation on Cantor-like Sets 4 If one of the above values is 0 rather than an m i the calculation is trivial. If M is generated by more than one element, by Furstenberg s result (967) that the only infinite closed subset of R/Z invariant under M is R/Z itself, C = I. Thus C is invariant under all integers, i.e. M = N, so m q m q a n = Q and Dirichlet s original function is recovered. If M has a single generator, denote it b 0, and then b = b r 0 for some r, and a = a r 0. In this case C is also the set constructed in the corresonding way for b 0, and some S 0 (which can be defined as the set satisfying that when written as base b 0 digits, S = S0, r the r-fold concatenations of elements of S 0.) Then m Q b an 0. Since m q m q = b k 0(b d 0 ) for some integers k, d, following (2) it suffices to observe that = b k+d 0 x b k 0x. Thus m q m q < b an 0 = b Qd 0. m q m q C, and Corollary 2. For all x C, there exist infinitely many solutions N, q N, q C to x < q q(log b0 q). /d Notice that the aroximation function s asymtotic behavior gets better as d 0, even though the first such q whose existence we rove can tend to infinity as b does. 3 Further investigation The obvious next ste would be to check whether this is the right rate of aroximation for C. For this urose, we make the following definitions: whereas the usual set BA consists of all reals x such that q x > c(x) q 2 /q Q, we define, relative to the aroximation function roven above, the intrinsic BA numbers to be all x C satisfying q x c(x) > /q Q C, q(log b0 q) /d and in comarison to the classical VWA definition q x < q 2+ɛ(x) for infinitely many /q, the intrinsic VWA numbers are those in C with q x < q +ɛ(x)

5 Intrinsic Aroximation on Cantor-like Sets 5 for infinitely many /q C instead. Then, one could say this function is correct if one roved that intrinsic BA is nonemty (and erhas of full dimension), or that intrinsic VWA has zero Hausdorff measure. These results aear difficult to achieve, however, without having a deeer understanding of the rationals in C. For the real line, there exists a large arsenal of useful information regarding distribution roerties of rationals, their quantity within bounds on q, etc., whereas for C nothing is even known about which denominators can aear - in reduced form, of course; the exression (2) is not useful here since it is not reduced. In fact Mahler oints out basically the same fundamental difficulty. To obtain the desired results, we believe a major new iece of information such as knowing exactly which denominators aear in C within given bounds (i.e. their density in the same sense as the classical number-theoretical study of the density of the rimes), or knowing how these rationals reel each other as a function of q, will be necessary. Until then, however, we make the following conjecture (which seems to be suorted by our exerimentation with some comuter data): Conjecture. Let Φ(i, j) = #{ q C, gcd(, q) =, i q j}. Then Φ(b n 0, b n+ 0 ) = O(a (+ɛ)n 0 ), for all ɛ > 0. This would imly that Intrinsic VWA has zero d-hausdorff measure. This set is equivalent to the one defined by a similar inequality with a (log b0 q) /d factor in the denominator, so the definition is analogous to the traditional one in that we simly divide our aroximation function by q ɛ. We remove the logarithmic factor for simlicity.

6 Intrinsic Aroximation on Cantor-like Sets 6 4 References L. Fishman, Schmidt s game on fractals, Israel J. Math. 7 (2009), no., H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a roblem in Diohantine aroximation, Math. Systems Theory (967), -49. A. Ghosh, A. Gorodnik, and A. Nevo, Diohantine aroximation and automorhic sectrum, Prerint, arxiv: v. D. Kleinbock, E. Lindenstrauss and B. Weiss, On fractal measures and diohantine aroximation, Selecta Math. 0 (2004), S. Kristensen, R. Thorn, and S.L. Velani, Diohantine aroximation and badly aroximable sets, Advances in Math. 203 (2006), D. Kleinbock and B. Weiss, Badly aroximable vectors on fractals, Israel Journal of Mathematics, 49 (2005), K. Mahler, Some suggestions for further research, Bulletin of the Australian Mathematical Society, 29 (984), W.M. Schmidt, Diohantine aroximation, Lecture Notes in Mathematics, vol. 785, Sringer-Verlag, Berlin, 980. B. Weiss, Almost no oints on a Cantor set are very well aroximable, Proc. R. Soc. Lond. A 457 (200),