Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

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1 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension Jack H. Lutz 1 and Neil Lutz 2 1 Depatment of Compute Science, Iowa State Univesity, Ames, IA, USA lutz@cs.iastate.edu 2 Depatment of Compute Science, Rutges Univesity, Piscataway, NJ, USA njlutz@utges.edu Abstact We fomulate the conditional Kolmogoov complexity of x given y at pecision, whee x and y ae points in Euclidean spaces and is a natual numbe. We demonstate the utility of this notion in two ways. 1. We pove a point-to-set pinciple that enables one to use the (elativized, constuctive) dimension of a single point in a set E in a Euclidean space to establish a lowe bound on the (classical) Hausdoff dimension of E. We then use this pinciple, togethe with conditional Kolmogoov complexity in Euclidean spaces, to give a new poof of the known, two-dimensional case of the Kakeya conjectue. This theoem of geometic measue theoy, poved by Davies in 1971, says that evey plane set containing a unit line segment in evey diection has Hausdoff dimension We use conditional Kolmogoov complexity in Euclidean spaces to develop the lowe and uppe conditional dimensions dim(x y) and Dim(x y) of x given y, whee x and y ae points in Euclidean spaces. Intuitively these ae the lowe and uppe asymptotic algoithmic infomation densities of x conditioned on the infomation in y. We pove that these conditional dimensions ae obust and that they have the coect infomation-theoetic elationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x : y) and Mdim(x : y) ACM Subject Classification F.1.3 Complexity Measues and Classes Keywods and phases algoithmic andomness, conditional dimension, geometic measue theoy, Kakeya sets, Kolmogoov complexity Digital Object Identifie /LIPIcs.STACS Intoduction This pape concens the fine-scale geomety of algoithmic infomation in Euclidean spaces. It shows how new ideas in algoithmic infomation theoy can shed new light on old poblems in geometic measue theoy. This intoduction explains these new ideas, a geneal pinciple fo applying these ideas to classical poblems, and an example of such an application. It also descibes a newe concept in algoithmic infomation theoy that aises natually fom this wok. Reseach suppoted in pat by National Science Foundation Gants and This wok was conducted at DIMACS and at Hebew Univesity. It was patially enabled though suppot fom the National Science Foundation unde gants CCF and CCF Jack H. Lutz and Neil Lutz; licensed unde Ceative Commons License CC-BY 34th Symposium on Theoetical Aspects of Compute Science (STACS 2017). Editos: Heibet Vollme and Bigitte Vallée; Aticle No. 53; pp. 53:1 53:13 Leibniz Intenational Poceedings in Infomatics Schloss Dagstuhl Leibniz-Zentum fü Infomatik, Dagstuhl Publishing, Gemany

2 53:2 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension Roughly fifteen yeas afte the mid-twentieth centuy development of the Shannon infomation theoy of pobability spaces [29], Kolmogoov ecognized that Tuing s mathematical theoy of computation could be used to efine the Shannon theoy to enable the amount of infomation in individual data objects to be quantified [17]. The esulting theoy of Kolmogoov complexity, o algoithmic infomation theoy, is now a lage entepise with many applications in compute science, mathematics, and othe sciences [20]. Kolmogoov poved the fist vesion of the fundamental elationship between the Shannon and algoithmic theoies of infomation in [17], and this elationship was made exquisitely pecise by Levin s coding theoem [18, 19]. (Solomonoff and Chaitin independently developed Kolmogoov complexity at aound the same time as Kolmogoov with somewhat diffeent motivations [30, 6, 7].) At the tun of the pesent centuy, the fist autho ecognized that Hausdoff s 1919 theoy of factal dimension [16] is an olde theoy of infomation that can also be efined using Tuing s mathematical theoy of computation, theeby enabling the density of infomation in individual infinite data objects, such as infinite binay sequences o points in Euclidean spaces, to be quantified [21, 22]. The esulting theoy of effective factal dimensions is now an active entepise with a gowing aay of applications [11]. The pape [22] poved a elationship between effective factal dimensions and Kolmogoov complexity that is as pecise as and uses Levin s coding theoem. Most of the wok on effective factal dimensions to date has concened the (constuctive) dimension dim(x) and the dual stong (constuctive) dimension Dim(x) [1] of an infinite data object x, which fo puposes of the pesent pape is a point in a Euclidean space R n fo some positive intege n. 1 The inequalities 0 dim(x) Dim(x) n hold geneally, with, fo example, Dim(x) = 0 fo points x that ae computable and dim(x) = n fo points that ae algoithmically andom in the sense of Matin-Löf [Mat66]. How can the dimensions of individual points dimensions that ae defined using the theoy of computing have any beaing on classical poblems of geometic measue theoy? The poblems that we have in mind hee ae poblems in which one seeks to establish lowe bounds on the classical Hausdoff dimensions dim H (E) (o othe factal dimensions) of sets E in Euclidean spaces. Such poblems involve global popeties of sets and make no mention of algoithms. The key to bidging this gap is elativization. Specifically, we pove hee a point-to-set pinciple saying that, in ode to pove a lowe bound dim H (E) α, it suffices to show that, fo evey A N and evey ε > 0, thee is a point x E such that dim A (x) α ε, whee dim A (x) is the dimension of x elative to the oacle A. We also pove the analogous point-to-set pinciple fo the classical packing dimension dim P (E) and the elativized stong dimension Dim A (x). We illustate the powe of the point-to-set pinciple by using it to give a new poof of a known theoem in geometic measue theoy. A Kakeya set in a Euclidean space R n is a set K R n that contains a unit line segment in evey diection. Besicovitch [2, 3] poved that Kakeya sets can have Lebesgue measue 0 and asked whethe Kakeya sets in the Euclidean plane can have dimension less than 2 [9]. The famous Kakeya conjectue assets a negative answe to this and to the analogous question in highe dimensions, i.e., states that evey 1 These constuctive dimensions ae Σ 0 1 effectivizations of Hausdoff and packing dimensions [13]. Othe effectivizations, e.g., computable dimensions, polynomial time dimensions, and finite-state dimensions, have been investigated, but only the constuctive dimensions ae discussed hee.

3 J. H. Lutz and N. Lutz 53:3 Kakeya set in a Euclidean space R n has Hausdoff dimension n. 2 This conjectue holds tivially fo n = 1 and was poven by Davies [9] fo n = 2. A vesion of the conjectue in finite fields has been poven by Dvi [12]. Fo Euclidean spaces of dimension n 3, it is an impotant open poblem with deep connections to othe poblems in analysis [35, 32]. In this pape we use ou point-to-set pinciple to give a new poof of Davies s theoem. This poof does not esemble the classical poof, which is not difficult but elies on Mastand s pojection theoem [26] and point-line duality. Instead of analyzing the set K globally, ou poof focuses on the infomation content of a single, judiciously chosen point in K. Given a Kakeya set K R 2 and an oacle A N, we fist choose a paticula line segment L K and a paticula point (x, mx+b) L, whee y = mx+b is the equation of the line containing L. 3 We then show that dim A (x, mx + b) 2. By ou point-to-set pinciple this implies that dim H (K) 2. Ou poof that dim A (x, mx + b) 2 equies us to fomulate a concept of conditional Kolmogoov complexity in Euclidean spaces. Specifically, fo points x R m and y R n and natual numbes, we develop the conditional Kolmogoov complexity K (x y) of x given y at pecision. This is a conditional vesion of the Kolmogoov complexity K (x) of x at pecision that has been used in seveal ecent papes (e.g., [24, 4, 15]). In addition to enabling ou new poof of Davies s theoem, conditional Kolmogoov complexity in Euclidean spaces enables us to fill a gap in effective dimension theoy. The fundamental quantities in Shannon infomation theoy ae the entopy (infomation content) H(X) of a pobability space X, the conditional entopy H(X Y ) of a pobability space X given a pobability space Y, and the mutual infomation (shaed infomation) I(X; Y ) between two pobability spaces X and Y [8]. The analogous quantities in Kolmogoov complexity theoy ae the Kolmogoov complexity K(u) of a finite data object u, the conditional Kolmogoov complexity K(u v) of a finite data object u given a finite data object v, and the algoithmic mutual infomation I(u : v) between two finite data objects u and v [20]. The above-descibed dimensions dim(x) and Dim(x) of a point x in Euclidean space (o an infinite sequence x ove a finite alphabet) ae analogous by limit theoems [27, 1] to K(u) and hence to H(X). Case and the fist autho have ecently developed and investigated the mutual dimension mdim(x : y) and the dual stong mutual dimension Mdim(x : y), which ae densities of the algoithmic infomation shaed by points x and y in Euclidean spaces [4] o sequences x and y ove a finite alphabet [5]. These mutual dimensions ae analogous to I(u : v) and I(X; Y ). What is conspicuously missing fom the above account is a notion of conditional dimension. In this pape we emedy this by using conditional Kolmogoov complexity in Euclidean space to develop the conditional dimension dim(x y) of x given y and its dual, the conditional stong dimension Dim(x y) of x given y, whee x and y ae points in Euclidean spaces. We pove that these conditional dimensions ae well behaved and that they have the coect infomation theoetic elationships with the peviously defined dimensions and mutual dimensions. The oiginal plan of ou poof of Davies s theoem used conditional dimensions, and we developed thei basic theoy to that end. Ou final poof of Davies s theoem does not use them, but conditional dimensions (like the conditional entopy and conditional Kolmogoov complexity that motivate them) ae vey likely to be useful in futue investigations. 2 Statements of the Kakeya conjectue vay in the liteatue. Fo example, the set is sometimes equied to be compact o Boel, and the dimension used may be Minkowski instead of Hausdoff. Since the Hausdoff dimension of a set is neve geate than its Minkowski dimension, ou fomulation is at least as stong as those vaiations. 3 One might naïvely expect that fo independently andom m and x, the point (x, mx + b) must be andom. In fact, in evey diection thee is a line that contains no andom point [23]. S TAC S

4 53:4 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension The est of this pape is oganized as follows. Section 2 biefly eviews the dimensions of points in Euclidean spaces. Section 3 pesents the point-to-set pinciples that enable us to use dimensions of individual points to pove lowe bounds on classical factal dimensions. Section 4 develops conditional Kolmogoov complexity in Euclidean spaces. Section 5 uses the peceding two sections to give ou new poof of Davies s theoem. Section 6 uses Section 4 to develop conditional dimensions in Euclidean spaces. 2 Dimensions of Points in Euclidean Spaces This section eviews the constuctive notions of dimension and mutual dimension in Euclidean spaces. The pesentation hee is in tems of Kolmogoov complexity. Biefly, the conditional Kolmogoov complexity K(w v) of a sting w {0, 1} given a sting v {0, 1} is the minimum length π of a binay sting π fo which U(π, v) = w, whee U is a fixed univesal self-delimiting Tuing machine. The Kolmogoov complexity of w is K(w λ), whee λ is the empty sting. We wite U(π) fo U(π, λ). When U(π) = w, the sting π is called a pogam fo w. The quantity K(w) is also called the algoithmic infomation content of w. Routine coding extends this definition fom {0, 1} to othe discete domains, so that the Kolmogoov complexities of natual numbes, ational numbes, tuples of these, etc., ae well defined up to additive constants. Detailed discussions of self-delimiting Tuing machines and Kolmogoov complexity appea in the books [20, 28, 11] and many papes. The definition of K(q) fo ational points q in Euclidean space is lifted in two steps to define the dimensions of abitay points in Euclidean space. Fist, fo x R n and N, the Kolmogoov complexity of x at pecision is K (x) = min{k(q) : q Q n B 2 (x)}, (2.1) whee B 2 (x) is the open ball with adius 2 and cente x. Second, fo x R n, the dimension and stong dimension of x ae dim(x) = lim inf K (x) and Dim(x) = lim sup K (x), (2.2) espectively. 4 Intuitively, dim(x) and Dim(x) ae the lowe and uppe asymptotic densities of the algoithmic infomation in x. These quantities wee fist defined in Canto spaces using betting stategies called gales and shown to be constuctive vesions of classical Hausdoff and packing dimension, espectively [22, 1]. These definitions wee explicitly extended to Euclidean spaces in [24], whee the identities (2.2) wee poven as a theoem. Hee it is convenient to use these identities as definitions. Fo x R n, it is easy to see that 0 dim(x) Dim(x) n, and it is known that, fo any two eals 0 α β n, thee exist uncountably many points x R n satisfying dim(x) = α and Dim(x) = β [1]. Applications of these dimensions in Euclidean spaces appea in [24, 14, 25, 10, 15]. 4 We note that K (x) = K(x ) + o(), whee x is the binay expansion of x, tuncated bits to the ight of the binay point. Howeve, it has been known since Tuing s famous coection [33] that binay notation is not a suitable epesentation fo the aguments and values of computable functions on the eals. (See also [34].) Hence, in ode to make ou definitions useful fo futhe wok in computable analysis, we fomulate complexities and dimensions in tems of ational appoximations, both hee and late.

5 J. H. Lutz and N. Lutz 53:5 3 Fom Points to Sets The cental message of this pape is a useful point-to-set pinciple by which the existence of a single high-dimensional point in a set E R n implies that the set E has high dimension. To fomulate this pinciple we use elativization. All the algoithmic infomation concepts in Sections 2 and 6 above can be elativized to an abitay oacle A N by giving the Tuing machine in thei definitions oacle access to A. Relativized Kolmogoov complexity K A (x) and elativized dimensions dim A (x) and Dim A (x) ae thus well defined. Moeove, the esults of Section 2 hold elative to any oacle A. We fist establish the point-to-set pinciple fo Hausdoff dimension. Let E R n. Fo δ > 0, define U δ (E) to be the collection of all countable coves of E by sets of positive diamete at most δ. That is, fo evey cove {U i } i N U δ (E), we have E i N U i and U i (0, δ] fo all i N, whee fo X R n, X = sup p,q X p q. Fo s 0, define { } Hδ s (E) = inf U i s : {U i } i N U δ (E). i N Then the s-dimensional Hausdoff oute measue of E is H s (E) = lim δ 0 + Hs δ (E), and the Hausdoff dimension of E is dim H (E) = inf {s > 0 : H s (E) = 0}. Moe details may be found in standad texts, e.g., [31, 13]. Theoem 1. (Point-to-set pinciple fo Hausdoff dimension) Fo evey set E R n, dim H (E) = min sup dim A (x). A N x E Thee things should be noted about this pinciple. Fist, while the left-hand side is the classical Hausdoff dimension, which is a global popety of E that does not involve the theoy of computing, the ight-hand side is a pointwise popety of the set that makes essential use of elativized algoithmic infomation theoy. Second, as the poof shows, the ight-hand side is a minimum, not meely an infimum. Thid, and most cucially, this pinciple implies that, in ode to pove a lowe bound dim H (E) α, it suffices to show that, fo evey A N and evey ε > 0, thee is a point x E such that dim A (x) α ε. 5 Fo the ( ) diection of this pinciple, we constuct the minimizing oacle A. The oacle encodes, fo a caefully chosen sequence of inceasingly efined coves fo E, the appoximate locations and diametes of all cove elements. Using this oacle, a point x R n can be appoximated by specifying an appopiately small cove element that it belongs to, which equies an amount of infomation that depends on the numbe of similaly-sized cove elements. We use the definition of Hausdoff dimension to bound that numbe. The ( ) diection can be shown using esults fom [24], but in the inteest of self-containment we pove it diectly. 5 The ε hee is useful in geneal but is not needed in some cases, including ou poof of Theoem 5 below. S TAC S

6 53:6 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension Poof of Theoem 1. Let E R n, and let d = dim H (E). Fo evey s > d we have H s (E) = 0, so thee is a sequence {{U t,s i } i N } t N of countable coves of E such that U t,s i 2 t fo evey i, t N, and fo evey sufficiently lage t we have U t,s s < 1. (3.1) i N i Let D = N 3 (Q (d, )). Ou oacle A encodes functions f A : D Q n and g A : D Q such that fo evey (i, t,, s) D, we have f A (i, t,, s) B 2 1(u) fo some u U t,s i and g A (i, t,, s) U t,s < 2 4. (3.2) i We will show, fo evey x E and ational s > d, that dim A (x) s. Fix x E and s Q (d, ). If fo any i 0, t 0 N we have x U t0,s i 0 and U t 0,s i 0 = 0, then U t0,s i 0 = {x}, so f A (i 0, t 0,, s) B 2 (x) fo evey N. In this case, let M be a pefix Tuing machine with oacle access to A such that, wheneve U(ι) = i N, U(τ) = t N, U(ρ) = N, and U(σ) = q Q (d, ), M(ιτρσ) = f A (i, t,, q). Now fo any N, let ι, τ, ρ, and σ be witnesses to K(i 0 ), K(t 0 ), K(), and K(s), espectively. Since i 0, t 0, and s ae all constant in and ρ = o(), we have ιτρσ = o(). Thus K A (x) = o(), and dim A (x) = 0. Hence assume that evey cove element containing x has positive diamete. be some cove element containing x. Let M be a self-delimiting Tuing machine with oacle access to A such that wheneve U(κ) = k N, U(τ) = l N, U(ρ) = N, and U(σ) = q Q (d, ), Fix sufficiently lage t, and let U t,s i x M (κτρσ) = f A (p, l,, q), whee p is the k th index i such that g A (i, t,, q) 2 3. Now fix t 1 such that U t,s i x [ 2 2, 2 1). Notice that g A (i x, t,, s) 2 3. Hence thee is some k such that, letting κ, τ, ρ, and σ be witnesses to K(k), K(t), K(), and K(s), espectively, M (κτρσ) B 2 1(u), fo some u U t,s i x. Because U t,s < 2 1 and x U t,s, we have Thus M (κτρσ) B 2 (x). K A (x) K(k) + K(t) + K(s) + K() + c, i x whee c is a machine constant fo M. Since s is constant in and t <, this expession is K(k) + o() log(k) + o(). By (3.1), thee ae fewe than 2 (+4)s indices i N such that U t,s 2 4, i i x

7 J. H. Lutz and N. Lutz 53:7 hence by (3.2) thee ae fewe than 2 (+4)s indices i N such that g A (i, t,, s) 2 3, so log(k) < ( + 4)s. Theefoe K A (x) s + o(). Thee ae infinitely many such, which can be seen by eplacing t above with + 2. We have shown dim A (x) = lim inf K A (x) s, fo evey ational s > d, hence dim A (x) d. It follows that min sup dim A (x) d. A N x E Fo the othe diection, assume fo contadiction that thee is some oacle A and d < d such that sup dim A (x) = d. x E Then fo evey x E, dim A (x) d. Let s (d, d). Fo evey N, define the sets and B = { B 2 (q) : q Q and K A (q) s } W = B k. k= Thee ae at most 2 ks+1 balls in each B k, so fo evey N and s (s, d), W W W s = W s k= W B k 2 ks+1 (2 1 k ) s k= = 2 1+s 2 (s s )k, k= which appoaches 0 as. As evey W is a cove fo E, we have H s (E) = 0, so dim H (E) s < d, a contadiction. The packing dimension dim P (E) of a set E R n, defined in standad texts, e.g., [13], is a dual of Hausdoff dimension satisfying dim P (E) dim H (E), with equality fo vey egula sets E. We also have the following. Theoem 2. (Point-to-set pinciple fo packing dimension) Fo evey set E R n, dim P (E) = min sup Dim A (x). A N x E 4 Conditional Kolmogoov Complexity in Euclidean Spaces We now develop the conditional Kolmogoov complexity in Euclidean spaces. S TAC S

8 53:8 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension Fo x R m, q Q n, and N, the conditional Kolmogoov complexity of x at pecision given q is ˆK (x q) = min {K(p q) : p Q m B 2 (x)}. (4.1) Fo x R m, y R n, and, s N, the conditional Kolmogoov complexity of x at pecision given y at pecision s is K,s (x y) = max { ˆK (x q) : q Q n B 2 s(y) }. (4.2) Intuitively, the maximizing agument q is the point nea y that is least helpful in the task of appoximating x. Note that K,s (x y) is finite, because ˆK (x q) K (x) + O(1). Fo x R m, y R n, and N, the conditional Kolmogoov complexity of x given y at pecision is K (x y) = K, (x y). (4.3) Theoem 3 (Chain ule fo K ). Fo all x R m and y R n, K (x, y) = K (x y) + K (y) + o(). We also conside the Kolmogoov complexity of x R m at pecision elative to y R n. Let K y (x) denote K Ay (x), whee A y N encodes the binay expansions of y s coodinates. The following lemma eflects the intuition that oacle access to y is at least as useful as any bounded-pecision estimate fo y. Lemma 4. Fo each m, n N thee is a constant c N such that, fo all x R m, y R n, and, s N, K y (x) K,s (x y) + K(s) + c. In paticula, K y (x) K (x y) + K() + c. 5 Kakeya Sets in the Plane This section uses the esults of the peceding two sections to give a new poof of the following classical theoem. Recall that a Kakeya set in R n is a set containing a unit line segment in evey diection. Theoem 5 (Davies [9]). Evey Kakeya set in R 2 has Hausdoff dimension 2. Ou new poof of Theoem 5 uses a elativized vesion of the following lemma. Lemma 6. Let m [0, 1] and b R. Then fo almost evey x [0, 1], lim inf K (m, b, x) K (b m) dim(x, mx + b). (5.1) Poof. We build a pogam that takes as input a pecision level, an appoximation p of x, an appoximation q of mx + b, a pogam π that will appoximate b given an appoximation fo m, and a natual numbe h. In paallel, the pogam consides each multiple of 2 in [0,1] as a possible appoximate value u fo the slope m, and it checks whethe each such u is consistent with the pogam s inputs. If u is close to m, then π(u) will be close to b, so up + π(u) will be close to mx + b. Any u that satisfies this condition is consideed a candidate fo appoximating m.

9 J. H. Lutz and N. Lutz 53:9 Some of these candidates may be false positives, in that thee can be values of u that ae fa fom m but fo which up + π(u) is still close to mx + b. Thus the pogam is also given an input h so that it can choose the coect candidate; it selects the h th candidate that aises in its execution. We will show that this h is often not lage enough to significantly affect the total input length. Fomally, let M be a Tuing machine that uns the following algoithm on input ρπση wheneve U(ρ) = N, U(η) = h N, and U(σ) = (p, q) Q 2 : candidate := 0 fo i = 0, 1,..., 2, in paallel: u i := 2 i v i := U(π, u i ) do atomically: if v i R and u i p + v i q < 2 2, then candidate := candidate + 1 if candidate = h, then etun (u i, v i, p) and halt Fix m [0, 1] and b R. Fo each N, let m = 2 m 2, and fix π testifying to the value of ˆK (b m ) and σ testifying to the value of K (x, mx + b). The poof is completed by the fou following claims. Intuitively, Claim 7 says that no point in B 2 (m) gives much less infomation about b than m does. Claim 8 states that thee is always some value of h that causes this machine to etun the desied output. Claim 9 says that fo almost evey x, this value does not gow too quickly with, and Claim 10 says that (5.1) holds fo evey such x. Claim 7. Fo evey N, K (b m) = ˆK (b m ) + o(). Claim 8. Fo each x [0, 1] and N, thee exists an h N such that M(ρπ σ η) B 2 1 (m, b, x), whee U(ρ) = and U(η) = h. Fo evey x [0, 1] and N, define h(x, ) to be the minimal h satisfying the conditions of Claim 8. Claim 9. Fo almost evey x [0, 1], log(h(x, )) = o(). Claim 10. Fo evey x [0, 1], if log(h(x, )) = o(), then lim inf K (m, b, x) K (b m) dim(x, mx + b). The lemma follows immediately fom Claims 9 and 10. Poof of Theoem 5. Let K be a Kakeya set in R 2. By Theoem 1, thee exists an oacle A such that dim H (K) = sup p K dim A (p). Let m [0, 1] such that dim A (m) = 1; such an m exists by Theoem 4.5 of [22]. K contains a unit line segment L of slope m. Let (x 0, y 0 ) be the left endpoint of such a segment. Let q Q [x 0, x 0 + 1/8], and let L be the unit segment of slope m whose left endpoint is (x 0 q, y 0 ). Let b = y 1 + qm, the y-intecept of L. By a elativized vesion of Lemma 6, thee is some x [0, 1/2] such that dim A,m,b (x) = 1 and lim inf K A (m, b, x) K A (b m) dim A (x, mx + b). S TAC S

10 53:10 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension (This holds because almost evey x [0, 1/2] is algoithmically andom elative to (A, m, b) and hence satisfies dim A,m,b (x) = 1.) Fix such an x, and notice that (x, mx + b) L. Now applying a elativized vesion of Theoem 3, dim A (x, mx + b) lim inf = lim inf = lim inf lim inf K A (m, b, x) K A (b m) K A (m, b, x) K A (b, m) + K A (m) K A (x b, m) + K A (m) K A (x b, m) K A (m) + lim inf. By Lemma 4, K A (x b, m) K A,b,m (x) + o(), so we have dim A (x, mx + b) lim inf K A,b,m (x) + lim inf = dim A,b,m (x) + dim A (m), which is 2 by ou choices of m and x. Since dim A (x, mx + b) = dim A (x + q, mx + b), K A (m) thee exists a point (x + q, mx + b) K such that dim A (x + q, mx + b) 2. By Theoem 1, the point-to-set pinciple fo Hausdoff dimension, this completes the poof. It is natual to ask what pevents us fom extending this poof to highe-dimensional Euclidean spaces. The point of failue in a diect extension would be Claim 9 in the poof of Lemma 6. Speaking infomally, the poblem is that the total numbe of candidates may gow as 2 (n 1), meaning that log(h(x, )) could be Ω((n 2)) fo evey x. 6 Conditional Dimensions in Euclidean Spaces The esults of Section 4, which wee used in the poof of Theoem 5, also enable us to give obust fomulations of conditional dimensions. Fo x R m and y R n, the lowe and uppe conditional dimensions of x given y ae dim(x y) = lim inf K (x y) and Dim(x y) = lim sup K (x y), (6.1) espectively. The use of the same pecision bound fo both x and y in (4.3) makes the definitions (6.1) appea abitay and bittle. The following theoem shows that this is not the case. Theoem 11. Let s : N N. If s() = o(), then, fo all x R m and y R n, and dim(x y) = lim inf Dim(x y) = lim sup K,s() (x y) K,s() (x y),.

11 J. H. Lutz and N. Lutz 53:11 The est of this section is devoted to showing that ou conditional dimensions have the coect infomation theoetic elationships with the peviously developed dimensions and mutual dimensions. Mutual dimensions wee developed vey ecently, and Kolmogoov complexity was the stating point. The mutual (algoithmic) infomation between two stings u, v {0, 1} is I(u : v) = K(v) K(v u). Again, outine coding extends K(u v) and I(u : v) to othe discete domains. Discussions of K(u v), I(u : v), and the coespondence of K(u), K(u v), and I(u : v) with Shannon entopy, Shannon conditional entopy, and Shannon mutual infomation appea in [20]. In paallel with (2.1) and (2.2), Case and J. H. Lutz [4] lifted the definition of I(p : q) fo ational points p and q in Euclidean spaces in two steps to define the mutual dimensions between two abitay points in (possibly distinct) Euclidean spaces. Fist, fo x R m, y R n, and N, the mutual infomation between x and y at pecision is I (x : y) = min {I(p : q) : p B 2 (x) Q m and q B 2 (y) Q n }, (6.2) whee B 2 (x) and B 2 (y) ae the open balls of adius 2 about x and y in thei espective Euclidean spaces. Second, fo x R m and y R n, the lowe and uppe mutual dimensions between x and y ae mdim(x : y) = lim inf I (x : y) and Mdim(x : y) = lim sup I (x : y), (6.3) espectively. Useful popeties of these mutual dimensions, especially including data pocessing inequalities, appea in [4]. Lemma 12. Fo all x R m and y R n, I (x : y) = K (x) K (x y) + o(). The following bounds on mutual dimension follow fom Lemma 12. Theoem 13. Fo all x R m and y R n, the following hold. 1. mdim(x : y) dim(x) Dim(x y). 2. Mdim(x : y) Dim(x) dim(x y). Ou final theoem is easily deived fom Theoem 3. Theoem 14 (Chain ule fo dimension). Fo all x R m and y R n, dim(x) + dim(y x) dim(x, y) 7 Conclusion dim(x) + Dim(y x) Dim(x, y) Dim(x) + Dim(y x). This pape shows a new way in which theoetical compute science can be used to answe questions that may appea unelated to computation. We ae hopeful that ou new poof of Davies s theoem will open the way fo using constuctive factal dimensions to make new pogess in geometic measue theoy, and that conditional dimensions will be a useful component of the infomation theoetic appaatus fo studying dimension. S TAC S

12 53:12 Algoithmic Infomation, Plane Kakeya Sets, and Conditional Dimension Acknowledgments. We thank Eic Allende fo useful coections and thee anonymous eviewes of an ealie vesion of this wok fo helpful input egading pesentation. Refeences 1 Kishna B. Atheya, John M. Hitchcock, Jack H. Lutz, and Elvia Mayodomo. Effective stong dimension in algoithmic infomation and computational complexity. SIAM J. Comput., 37(3): , doi: /s A. S. Besicovitch. Su deux questions d intégabilité des fonctions. Jounal de la Société de physique et de mathematique de l Univesite de Pem, 2: , A. S. Besicovitch. On Kakeya s poblem and a simila one. Mathematische Zeitschift, 27: , Adam Case and Jack H. Lutz. Mutual dimension. ACM Tansactions on Computation Theoy, 7(3):12, Adam Case and Jack H. Lutz. Mutual dimension and andom sequences. In Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, editos, Mathematical Foundations of Compute Science th Intenational Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Poceedings, Pat II, volume 9235 of Lectue Notes in Compute Science, pages Spinge, Gegoy J. Chaitin. On the length of pogams fo computing finite binay sequences. J. ACM, 13(4): , Gegoy J. Chaitin. On the length of pogams fo computing finite binay sequences: statistical consideations. J. ACM, 16(1): , Thomas R. Cove and Joy A. Thomas. Elements of Infomation Theoy. Wiley, second edition, Roy O. Davies. Some emaks on the Kakeya poblem. Poc. Cambidge Phil. Soc., 69: , Randall Doughety, Jack H. Lutz, R. Daniel Mauldin, and Jason Teutsch. Tanslating the Canto set by a andom eal. Tansactions of the Ameican Mathematical Society, 366: , Rod Downey and Denis Hischfeldt. Algoithmic Randomness and Complexity. Spinge- Velag, Zeev Dvi. On the size of Kakeya sets in finite fields. J. Ame. Math. Soc., 22: , Kenneth Falcone. Factal Geomety: Mathematical Foundations and Applications. Wiley, thid edition, Xiaoyang Gu, Jack H. Lutz, and Elvia Mayodomo. Points on computable cuves. In FOCS, pages IEEE Compute Society, doi: /focs Xiaoyang Gu, Jack H. Lutz, Elvia Mayodomo, and Philippe Mose. Dimension specta of andom subfactals of self-simila factals. Ann. Pue Appl. Logic, 165(11): , Felix Hausdoff. Dimension und äussees Mass. Mathematische Annalen, 79: , Andei N. Kolmogoov. Thee appoaches to the quantitative definition of infomation. Poblems of Infomation Tansmission, 1(1):1 7, Leonid A. Levin. On the notion of a andom sequence. Soviet Math Dokl., 14(5): , Leonid A. Levin. Laws of infomation consevation (nongowth) and aspects of the foundation of pobability theoy. Poblemy Peedachi Infomatsii, 10(3):30 35, Ming Li and Paul M.B. Vitányi. An Intoduction to Kolmogoov Complexity and Its Applications. Spinge, thid edition, Jack H. Lutz. Dimension in complexity classes. SIAM J. Comput., 32(5): , 2003.

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