aa r X i v : . [ m a t h . D S ] A ug A NONLINEAR VERSION OF THE NEWHOUSETHICKNESS THEOREM
KAN JIANG
Abstract.
Let C and C be two Cantor sets with convex hull [0 , τ ( C ) · τ ( C ) ≥
1, then the arithmetic sum C + C is an interval, where τ ( C i ) , ≤ i ≤ C i . In this paper, we generalize this thickness theorem as follows. Let K i ⊂ R , i = 1 , · · · , d , be some Cantor sets (perfect and nowhere dense)with convex hull [0 , f ( x , · · · , x d − , z ) ∈ C is a continuousfunction defined on R d . Denote the continuous image of f by f ( K , · · · , K d ) = { f ( x , · · · x d − , z ) : x i ∈ K i , z ∈ K d , ≤ i ≤ d − } . If for any ( x , · · · , x d − , z ) ∈ [0 , d , we have( τ ( K i )) − ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i f∂ z f (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( K d ) , ≤ i ≤ d − f ( K , · · · , K d ) is a closed interval. We give two applications.Firstly, we partially answer some questions posed by Takahashi [16].Secondly, we obtain various nonlinear identities, associated with thecontinued fractions with restricted partial quotients, which can repre-sent real numbers. Introduction
Let K and K be two Cantor sets with convex hull [0 , τ ( K ) · τ ( K ) > τ ( K ) · τ ( K ) ≥ K + K is an interval, where τ ( K i ) denotes the thickness of K i , i = 1 , . The arithmetic sum of Cantorsets appears naturally in bifurcation theory. Palis [13] posed the followingproblem which is currently known as the Palis’ conjecture. Whether it istrue (at least generically) that the arithmetic sum of dynamically definedCantor sets either has measure zero or contains an interior. This conjecturewas solved by Moreira and Yoccoz [11]. The Newhouse’s thickness theoremis a very powerful result which can judge whether the arithmetic sum of twoCantor sets contains interior. Astels [1, Theorem 2.4] generalized the New-house’s thickness theorem by considering multiple sum of Cantor sets. Hemade use of this new thickness theorem to prove some identities which canrepresent real numbers. Astels’ thickness theorem implies many interestingresults. For instance, we may prove some Waring type result as follows, see
Date : August 21, 2020.2010
Mathematics Subject Classification.
Primary: 28A80, Secondary:11K55. [2, 17]. For each k ≥
2, there is a number n ( k ) ≤ k such that for any x ∈ [0 , n ( k )], we have x = n ( k ) X i =1 x ki , where x i is taken from the middle-third Cantor set. The Newhouse andAstels’ thickness theorems are very useful when we consider the sum of twoCantor sets. It is natural to consider a nonlinear version of Newhouse’sthickness theorem. Suppose f ( x , · · · , x d − , z ) ∈ C is a continuous functiondefined on R d . Denote the continuous image of f by f ( K , · · · , K d ) = { f ( x , · · · , x d − , z ) : x i ∈ K i , z ∈ K d , ≤ i ≤ d − } , where { K i } di =1 are general Cantor sets. To the best of our knowledge,there are very few results about f ( K , · · · , K d ). Generally, to considerthe topological structure of f ( K , · · · , K d ) is a difficult question. As weknow very little information about { K i } di =1 . Moreover, the nonlinearity of f ( x , · · · , x d − , z ) makes the abstract set f ( K , · · · , K d ) obscure. The mainaim of this paper is to give some sufficient conditions on f ( x , · · · , x d − , z )such that f ( K , · · · , K d ) is a closed interval.We now introduce some related results concerning with the continuousimage of f in R . The first one, to the best of our knowledge, is due toSteinhaus [15] who proved in 1917 the following interesting results: C + C = { x + y : x, y ∈ C } = [0 , , C − C = { x − y : x, y ∈ C } = [ − , , where C is the middle-third Cantor set. It is worth pointing out that Stein-haus also proved that for any two sets with positive Lebesgue measure, theirarithmetic sum contains interiors. In 2019, Athreya, Reznick and Tyson [2]proved that C ÷ C = (cid:26) xy : x, y ∈ C, y = 0 (cid:27) = ∞ [ n = −∞ (cid:20) − n , − n (cid:21) ∪ { } . In [8], Gu, Jiang, Xi and Zhao gave the topological structure of C · C = { xy : x, y ∈ C } . They proved that the exact Lebesgue measure of C · C is about 0 . . Wegive some remarks on the above results. The main idea of [2] is effectivefor homogeneous self-similar sets. For a general self-similar set or somegeneral Cantor set, we may not utilize their idea directly. Fraser, Howroydand Yu [6] studied the dimensions of sumsets and iterated sumsets, andprovided natural conditions which guarantee that a set F ⊂ R satisfiesdim B F + F > dim B F . The reader can find more related references in [6].For higher dimensions, namely R d , d ≥
3, there are relatively few results.Banakh, Jab lo´nska and Jab lo´nski [3] proved under some mild conditionsthat the arithmetic sum of d many compact connected sets in R d has non-empty interior. As a consequence, every compact connected set in R d not NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 3 lying a hyperplane is arithmetically thick. A compact set E ⊂ R d is said tobe arithmetically thick if there exists a positive integer n so that the n -foldarithmetic sum of E has non-empty interior. Recently Feng and Wu [7]defined the thickness of sets in R d , and proved the arithmetic thickness forseveral classes of fractal sets, including self-similar sets, self-conformal setsin R d (with d ≥
2) and some self-affine sets. All these elegant results areconcerning with arithmetic sum. They introduced some new ideas which arevery useful to analyze the sets in R d . In this paper, we consider similar problems. However, our main moti-vation is to generalize the Newhouse’s thickness theorem for some generalfunctions. Before we introduce the main results of this paper, we give somedefinitions. First, we give a well-known method that can generate a Can-tor set. For simplicity, we let I = [0 , , O . Then there are two closed intervalsleft, we denote them by B and B . Therefore, [0 ,
1] = B ∪ O ∪ B . Let E = B ∪ B . In the second level, let O and O be open intervals that aredeleted from B and B respectively, then we clearly have B = B ∪ O ∪ B , B = B ∪ O ∪ B . Let E = B ∪ B ∪ B ∪ B . Repeating this process, we can generate E n +1 from E n by removing anopen interval from each closed interval in the union which consists of E n .We assume that the deleted open intervals are arranged by the decreasinglengths, i.e. the lengths of deleted open intervals are decreasing. If for somelevels, the deleted open intervals have the same length, then we can deletethese open intervals in any order. To avoid triviality, we make the followingrule. Let B ω be a closed interval in some level, then we delete an openinterval O ω from B ω , i.e. B ω = B ω ∪ O ω ∪ B ω . We assume that the length of O ω is positive and strictly smaller than B ω .We let K = ∩ ∞ n =1 E n , and call K a Cantor set. The above rule is to ensure the Cantor set is perfectand nowhere dense.The next definition is the famous Newhouse’s thickness. Given a Cantorset K = ∩ ∞ n =1 E n . Let B ω be a closed interval in some level. Then by the construction of K ,we have B ω = B ω ∪ O ω ∪ B ω , K. JIANG where O ω is an open interval while B ω and B ω are closed intervals. Wecall B ω and B ω bridges of K , and O ω gap of K . Let τ ω ( B ω ) = min (cid:26) | B ω || O ω | , | B ω || O ω | (cid:27) , where | · | means length. We define the thickness of K by τ ( K ) = inf B ω τ ω ( B ω ) . Here the infimum takes over all bridges in every level.Now we state the main result of this paper.
Theorem 1.1.
Let { K i } di =1 be Cantor sets with convex hull [0 , . Supposethat f ( x , · · · , x d − , z ) ∈ C . If for any ( x , · · · , x d − , z ) ∈ [0 , d , we have ( τ ( K i )) − ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i f∂ z f (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( K d ) , ≤ i ≤ d − then f ( K , · · · , K d ) = H, where H = (cid:20) min ( x , ··· ,z ) ∈ K ×···× K d f ( x , · · · , z ) , max ( x , ··· ,z ) ∈ K ×···× K d f ( x , · · · , z ) (cid:21) , and τ ( K i ) , i = 1 , · · · , d, denotes the thickness of K i .Remark . This result partially generalizes [1, Theorem 2.4]. Theorem 1.1can be given an explanation from geometric measure theory. If the convexhull of each K i is different, then we need to assume each K i is not containedin any other K j ’s gaps, where j = i . As for this case f ( K , · · · , K d ) mayhave Lebesgue measure zero. When we use Theorem 1.1, we need to abideby this rule. Corollary 1.3.
Let { K i } di =1 be Cantor sets with convex hull [0 , . Supposethat f ( x , · · · x d − , z ) ∈ C . If for any ( x , · · · , x d − , z ) ∈ [0 , d , we have ( τ ( K i )) − ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i f∂ z f (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( K d ) , ≤ i ≤ d − then for any w ∈ H the hypersurface f ( x , · · · , x d − , z ) = w intersects with K × · · · × K d . For d = 2 we have the following result which can be viewed as a nonlinearversion of the Newhouse’s thickness theorem. Corollary 1.4.
Let K and K be two Cantor sets with convex hull [0 , .Suppose f ( x, y ) ∈ C . If for any ( x, y ) ∈ [0 , , we have ( τ ( K )) − ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f∂ y f (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( K ) , NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 5 then f ( K , K ) = (cid:20) min ( x,y ) ∈ K × K f ( x, y ) , max ( x,y ) ∈ K × K f ( x, y ) (cid:21) = H, where τ ( K i ) , i = 1 , denotes the thickness of K i . In particular, if we take alinear function f ( x, y ) = x + y, and τ ( K i ) ≥ , i = 1 , , then K + K is an interval.Remark . The conditions in Corollary 1.4 imply that τ ( K ) τ ( K ) ≥ τ ( K ) τ ( K ) < f ( K , K ) does not contain some interiors, see for instance inCorollary 1.7 and the remarks below. By the Newhouse’s thickness theorem,if τ ( K ) τ ( K ) ≥
1, then K + K is an interval. However, under the condition τ ( K ) τ ( K ) ≥
1, we may not have that K · K = { xy : x ∈ K , y ∈ K } is still an interval. A simple example is the middle-third Cantor set, denotedby C . The thickness of C is 1. We have C + C = [0 , . However, C · C ⊂ [0 , / ∪ [4 / , C · C is not an interval.Therefore, for a general f , if we want f ( K , K ) to be some interval, we mayexpect more strong conditions on f besides τ ( K ) τ ( K ) ≥ Corollary 1.6.
Let K and K be two Cantor sets with convex hull [0 , .If τ ( K ) τ ( K ) > , then there are uncountably many nonlinear functions f ( x, y ) ∈ C such that f ( K , K ) is an interval. The condition on partial derivatives in Theorem 1.1 can be weakenedwhen we consider some homogeneous self-similar sets. Indeed, the thicknessgives little information about the relation between gaps and bridges. Ifwe elaborately analyze their relation, we may obtain more delicate result.For instance, with a similar discussion as Theorem 1.1, we may prove thefollowing result.
Corollary 1.7.
Let K λ be the attractor of the IFS { f ( x ) = λx, f ( x ) = λx + 1 − λ, < λ < / } . Suppose that f ( x, y ) ∈ C is a continuous function defined on R . If for any ( x, y ) ∈ [0 , , we have − λλ ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f∂ y f (cid:12)(cid:12)(cid:12)(cid:12) ≤ − λ , K. JIANG then f ( K λ , K λ ) = (cid:20) min ( x,y ) ∈ K λ × K λ f ( x, y ) , max ( x,y ) ∈ K λ × K λ f ( x, y ) (cid:21) . Remark . The conditions in Corollary 1.7 imply that1 − λλ ≤ − λ , i.e. 1 / ≤ λ < / . This condition is natural as for any f ( x, y ) ∈ C and 0 < λ < /
4, we havedim H ( f ( K λ , K λ )) ≤ dim H ( K λ × K λ ) = 2 log 2 − log λ < . In other words, if 0 < λ < /
4, then f ( K λ , K λ ) cannot be an interval.Note that τ ( K λ ) = λ − λ < < λ < /
3. For this case, theNewhouse’s thickness theorem does not offer any information for f ( K λ , K λ ).Moreover, by [1, Theorem 2.4], γ ( K λ ) = τ ( K λ ) τ ( K λ ) + 1 = λ − λ , we cannotmake use of Astels’ result to consider whether f ( K λ , K λ ) is an interval asfor 1 / < λ < / γ ( K λ ) <
1. In fact, for the sum of two Cantorsets, the Newhouse’s thickness theorem and Astels’ thickness theorem areexactly the same. We mention some related work. In [14], Pourbarat provedunder some assumptions that g ( K λ ) + g ( K λ ) = { g ( x ) + g ( y ) : x ∈ K λ , y ∈ K λ } contains an interval, where g , g ∈ C . In Corollary 1.4, we prove undersome conditions that f ( K , K ) is an interval for general Cantor sets.In [16], Takahashi asked what is the topological structure of K λ · K λ .He also posed the question for the multiple product of some K λ i . In fact, wecan simultaneously consider multiplication and division on K λ . We partiallyanswer his questions as follows. Corollary 1.9.
Let { K λ i } di =1 be self-similar sets with < λ i < / , i =1 , · · · , d. If for any ≤ i ≤ d − − λ i λ i ≤ − λ d − λ i ≤ λ d − λ d , then Π di =1 K ǫ i λ i = { Π di =1 x ǫ i i : x i ∈ K λ i , ǫ i ∈ {− , } , x i = 0 if ǫ i = − } = U, where U = [ k ,k , ··· ,k d ∈ N λ ǫ k λ ǫ k · · · λ ǫ d k d d [ δ, η ] ∪ { } ,δ = Π ǫ i =1 (1 − λ i ) , η = Π ǫ i = − (1 − λ i ) − . NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 7
Remark . To avoid triviality, in the definition of Π di =1 K ǫ i λ i , we assumethat there exist some 1 ≤ i, j ≤ d such that ǫ i = − , ǫ j = 1. For this case,Π di =1 K ǫ i λ i contains 0. If ǫ i = − , for any 1 ≤ i ≤ d , then Π di =1 K ǫ i λ i does notcontain 0 . We may find more similar conditions, as in the above corollary, whichallow us to describe the structure of Π di =1 K ǫ i λ i . Note thatΠ di =1 K ǫ i λ i = [ k ,k , ··· ,k d ∈ N λ ǫ k λ ǫ k · · · λ ǫ d k d d (Π di =1 ( g K λ i ) ǫ i ) ∪ { } . where each g K λ i is the right similar copy of K i . The above result only investi-gates f ( x , · · · , x d ) on g K λ × · · · × g K λ d , see the details in the proof. Indeed,we may decompose each g K λ i into two sub self-similar sets, and analyze thepartial derivatives on these sub similar sets. We leave these considerationsto the reader.The following result indicates that the multiplication and division on someself-similar sets may simultaneously reach their maximal ranges. Corollary 1.11.
Let K be the attractor of the following IFS { f ( x ) = λ x, f ( x ) = λ x + 1 − λ , < λ ≤ λ < , λ + λ < } . Then the following conditions are equivalent: (1) K · K = { x · y : x, y ∈ K } = [0 , λ ≥ (1 − λ ) ;(3) K ÷ K = (cid:26) xy : x, y ∈ K, y = 0 (cid:27) = R . Finally, we give an application to the continued fractions with restrictedpartial quotients. We first give some basic definitions. Let m ∈ N ≥ . Define F ( m ) = { [ t, a , a , · · · ] : t ∈ Z , ≤ a i ≤ m for i ≥ } , where [ t, a , a , · · · ] = t + 1 a + 1 a + 1 · · · . For each l ∈ N + , define G ( l ) = { [ t, a , a , · · · ] : t ∈ Z , a i ≥ l for i ≥ }∪{ [ t, a , a , · · · , a k ] : t, k, ∈ Z , k ≥ , and a i ≥ l for 1 ≤ i ≤ k } . Generally, let B be a finite digits set, denote by F ( B ) the set of pointswhich have an infinite continued fraction expansions with all partial quo-tients, except possibly the first, members of B . When B is infinite, then we K. JIANG define F ( B ) in a similar way ( F ( B ) also includes some real numbers withfinite continued expansions). For more detailed introduction, see [1, 4]. Let F t ( B ) be a subset of F ( B ) with the first partial quotient t ∈ Z . Therefore, F ( B ) = ∪ t ∈ Z F t ( B ) . With this notation, we have F ( m ) = ∪ t ∈ Z F t ( B )where B = { , , · · · , m } . It is not difficult to calculate the Newhouse thick-ness of F t ( B ), see for instance [1, Lemma 4.3, Lemma 4.4].The main motivation why we consider continued fractions with restrictedpartial quotients is due to some well-known results. Hall [9] proved that F (4) + F (4) = { x + y : x, y ∈ F (4) } = R . Diviˇs [5] showed that Hall’s result is sharp in some sense as F (3) + F (3) = R . Here we use one simple fact, i.e. F ( n ) ⊂ F ( n + 1) for any n ≥
2. Hlavka[10] generalized Hall’s result and proved that F (3) + F (4) = R , F (2) + F (7) = R , F (2) + F (4) = R . Astels [1] showed F (2) ± F (5) = R , F (3) − F (4) = R , F (2) − F (4) = R , F (3) − F (3) = R . All of the above equations are linear, i.e. the associated function f ( x, y ) = x ± y is linear. It is natural to ask can we obtain similar results for some non-linearfunctions. Note that F t ( B ) , t ∈ Z is a Cantor set. Therefore, by Theorem1.1 and the thickness of F t ( B ) we can obtain some nonlinear results forthe arithmetic on F ( B ) if we appropriately choose the functions f ( x, y ).We only give the following equations. The reader may find more similaridentities which can represent real numbers. Corollary 1.12. F (7) ± F (7) = R , ( C + 1) + 2 F (6) = R , f ( K , K , K ) = R , where K = K = C + 1 , K = F (6) , f ( x, y, z ) = 0 . x + xy + z, and C is the middle-third Cantor set. This paper is arranged as follows. In Section 2, we prove the main resultsof this paper. In Section 3, we give some identities which can represent realnumbers. Finally, we give some remarks and questions.
NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 9 Proofs of main results
Proof of Corollary 1.4.
We first prove Corollary 1.4. The proof ofgeneral result, i.e. Theorem 1.1, depends on Corollary 1.4.Clearly, f ( K , K ) ⊂ H. To prove f ( K , K ) = H, we suppose on the contrary that f ( K , K ) = H ,then we shall find some contradictions. Therefore, we finish the proof ofCorollary 1.4.If f ( K , K ) = H, then we can find some z ∈ H such that Φ z does not intersect with K × K , where Φ z = { ( x, y ) ∈ [0 , : f ( x, y ) = z } . By virtue of the continuity of f , it follows that Φ z is a compact set. Notethat Φ z can be covered by countably many strips Γ of the form { ( x, y ) : x ∈ O, y ∈ [0 , } or { ( x, y ) : y ∈ O, x ∈ [0 , } , where O ’s are the deleted open intervals when we construct K i , i = 1 , z , we may find finitely many strips from the abovecovering, i.e. Φ z ⊂ ∪ ni =1 Γ i . By the construction of Cantor sets (we mainly use the fact that the deletedopen intervals are pairwise disjoint) and the continuity of f ( x, y ), it followsthat for any 1 ≤ i ≤ n −
1, Γ i is perpendicular to Γ i +1 .Suppose that Γ min is the strip which has minimal width (every strip haslength 1) among ∪ ni =1 Γ i . For every Γ i , ≤ i ≤ n , we denote its width by L i . Then we have the following lemma.
Lemma 2.1.
The strip Γ min does not parallel with the x -axis. Proof:
We prove this lemma for three cases. Firstly, if the strip Γ min isparallelling with the x -axis, and it is closest to the origin. Then by theimplicit function theorem and the minimal width of Γ min (we denote itswidth by L min ), there exists some ( x , y ) ∈ Γ min such that (cid:12)(cid:12)(cid:12)(cid:12) dydx | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f | ( x ,y ) ∂ y f | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) < L min ρ ≤ L min L τ ( K ) ≤ τ ( K ) , see the first graph of Figure 1. This contradicts to the condition in Corollary1.4. Secondly, if the strip Γ min is parallelling with the x -axis, and it is closestto the line y = 1, then we may find a similar contradiction as the first case.Finally, suppose the strip Γ min is parallelling with the x -axis, and there isat least one parallelling strip below and above Γ min , respectively. Let Γ and Γ be two strips that are perpendicular to Γ min such that Φ z enters andleaves the Γ min . The entrance point is in Γ while the leaving point is in Γ . xy o f ( x, y ) = zL min L ρA BCD xy oL min L L , A BCD L f ( x, y ) = z xy o L min L L L , f ( x, y ) = z Figure 1.
Let L , be the distance between Γ and Γ . Then by the implicit functiontheorem there exists some ( x , y ) ∈ Γ min such that (cid:12)(cid:12)(cid:12)(cid:12) dydx | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f | ( x ,y ) ∂ y f | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) < L min L , ≤ L min min { L , L } τ ( K ) ≤ τ ( K ) , see the second graph of Figure 1. This contradicts to the assumption ofCorollary 1.4. Hence, we have proved Lemma 2.1.Similarly, we can prove the following lemma. Lemma 2.2.
The strip Γ min does not parallel with the y -axis. Proof:
Suppose that Γ min is parallelling with the y -axis. We prove thislemma in three cases which are similar to Lemma 2.1. For simplicity, weonly prove the following case.Suppose there is at least one parallelling strip located on the left andright of Γ min , respectively. Then we let Γ and Γ be two strips that areperpendicular to Γ min such that the Φ z enters and leaves the Γ min . Theentrance point is in Γ and the leaving point is in Γ . Denote by L , thedistance between Γ and Γ , see the third graph of Figure 1. Therefore, bythe implicit function theorem again, there exists some ( x , y ) ∈ Γ min suchthat (cid:12)(cid:12)(cid:12)(cid:12) dydx | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f | ( x ,y ) ∂ y f | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) > L , L min ≥ min { L , L } τ ( K ) L min ≥ τ ( K ) . NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 11
This is a contradiction.
Proof of Corollary 1.4
Corollary 1.4 follows from Lemmas 2.1 and 2.2.2.2.
Proof of Theorem 1.1.
Now, we prove Theorem 1.1. The main ideais exactly the same as Corollary 1.4. Firstly, we clearly have f ( K , · · · , K d ) ⊂ H. If f ( K , · · · , K d ) ( H, then there exists some w ∈ H such that the hypersurface f ( x , · · · , z ) = w does not intersect with K × K × · · · × K d . Now we construct the following setΨ w = { ( x , x , · · · , z ) ∈ [0 , d : f ( x , · · · , z ) = w } . It is a compact set by the continuity of f . Hence, we can find finitely many d -dimensional cubes of the formΛ = ∆ × ∆ × · · · × ∆ d ⊂ [0 , d , such that there is a unique ∆ i ( [0 ,
1] is an open interval for some 1 ≤ i ≤ d ,and the rest ∆ j = [0 , , j = i . We call each ∆ i , ≤ i ≤ d an edge of∆ × ∆ × · · · × ∆ d . For simplicity, we call the edge which is not equalto [0 ,
1] the axis edge. Without loss of generality, we may assume thatΨ w ⊂ ∪ ni =1 Λ i , Λ i is perpendicular to Λ i +1 for 1 ≤ i ≤ n −
1, i.e. the axisedges of Λ i and Λ i +1 have different subscripts. Let Λ min be the cube withminimal length, i.e. one edge of Λ min has minimal length among ∪ ni =1 Λ i . Weshall prove that the above covering, i.e. ∪ ni =1 Λ i , does not exists. Therefore,we prove the desired result.Let Λ min = [0 , i − × ( p i , q i ) × [0 , d − i , ( p i , q i ) ( [0 , , ≤ i ≤ d. Suppose 1 ≤ i ≤ d −
1, for the hypersurface f ( x , · · · , z ) = w, we fix x j , j = i, d (we let x d = z ). Therefore the hypersurface f ( x , · · · , z ) = w can be covered by Ω i ∪ Ω d , whereΩ i = ∪{ x } × { x } × · · · × { x i − } × ( p i , q i ) × { x i +1 } × · · · × { x d − } × [0 , , andΩ d = ∪{ x } × { x } × · · · × { x i − } × [0 , × { x i +1 } × · · · × { x d − } × ( p d , q d ) . Here the unions in the above equations mean finite (by the compactness ofΨ w ) deleted open intervals when we construct K i and K d . Since we fix x j , j = i, d , it follows that the hypersurface f ( x , · · · , z ) = w becomes a curve on a plane which is a translation of the x i Oz plane. We letthis curve be Υ. By the above discussion, we have Υ ⊂ Ω i ∪ Ω d . Nevertheless,by Lemmas 2.1 and 2.2, Υ cannot be in Ω i ∪ Ω d , which is a contradiction.If i = d , then Λ min = [0 , d − × ( p d , q d ) . For this case, we can also prove similarly as above, and obtain a contradic-tion. Hence, we finish the proof.2.3.
Proof of Corollary 1.6.
Note that τ ( K ) τ ( K ) > ⇔ τ ( K ) < τ ( K ) . If τ ( K ) > τ ( K ) >
1, then there exist some α, β ∈ R + such that τ ( K ) > α, τ ( K ) > β. Now, we let f ( x, y ) = αx + βy + x + y . Since the convex hull of K i , i = 1 , , f ( K , K ) is an interval.If τ ( K ) > > τ ( K ), then we can find some γ, ζ ∈ R + such that1 τ ( K ) < γ ζ , γ + 2 ζ < τ ( K ) . We let f ( x, y ) = x + y + γx + ζy. It is easy to check the conditions in Corollary 1.4. Hence, f ( K , K ) is aninterval.2.4. Proof of Corollary 1.7.
The proof is almost the same as the proofof Corollary 1.4. We only need to prove Lemma 2.2 under the assumption (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f∂ y f (cid:12)(cid:12)(cid:12)(cid:12) ≤ − λ . For simplicity, we only prove the first case of Lemma 2.2. We still use theterminology of Lemma 2.2, and the third graph of Figure 1. By Lemma 2.1Γ min cannot parallel with x -axis. By the minimality of Γ min , we have λ min { L , L } ≥ L min . Therefore, by the implicit function theorem, there exists some ( x , y ) ∈ Γ min such that (cid:12)(cid:12)(cid:12)(cid:12) dydx | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f | ( x ,y ) ∂ y f | ( x ,y ) (cid:12)(cid:12)(cid:12)(cid:12) > L , L min ≥ min { L , L } τ ( K λ ) L min ≥ τ ( K λ ) λ = 11 − λ . This is a contradiction.
NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 13
Proof of Corollary 1.9.
Let f ( x , · · · , x d ) = Π di =1 x ǫ i i . It is easy tocheck that (cid:12)(cid:12)(cid:12)(cid:12) ∂ x i f∂ x d f (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) x d x i (cid:12)(cid:12)(cid:12)(cid:12) , ≤ i ≤ d − . Note thatΠ di =1 K ǫ i λ i = [ k ,k , ··· ,k d ∈ N λ ǫ k λ ǫ k · · · λ ǫ d k d d (Π di =1 ( g K λ i ) ǫ i ) ∪ { } , where g K λ i is the right similar copy of K i . Note that the convex hull of g K λ × g K λ × · · · × g K λ d is V = [1 − λ , × [1 − λ , × · · · × [1 − λ d , . Therefore, for any ( x , · · · , x d ) ∈ V , we have1 − λ d ≤ (cid:12)(cid:12)(cid:12)(cid:12) x d x i (cid:12)(cid:12)(cid:12)(cid:12) ≤ − λ i . Then by the following conditions − λ i λ i ≤ − λ d − λ i ≤ λ d − λ d , we clearly have τ ( K λ i ) − ≤ (cid:12)(cid:12)(cid:12)(cid:12) x d x i (cid:12)(cid:12)(cid:12)(cid:12) ≤ τ ( K λ d ) , ≤ i ≤ d − . Now, Corollary 1.9 follows from Theorem 1.1.2.6.
Proof of Corollary 1.11.
We first prove (1) ⇒ (2). This is clear as K ⊂ [0 , λ ] ∪ [1 − λ , ⇒ K · K ⊂ [0 , λ ] ∪ [(1 − λ ) , . Now, we prove that (2) ⇒ (1). Let f ( x, y ) = xy . First, we have the followingequation: K · K = ∪ ∞ i =0 λ i ( f ( K ) · f ( K )) ∪ { } . The convex hull of f ( K ) is [1 − λ , f on [1 − λ , . It is easy to calculate that1 − λ ≤ (cid:12)(cid:12)(cid:12)(cid:12) ∂ x f∂ y f (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) yx (cid:12)(cid:12)(cid:12) ≤ − λ for any ( x, y ) ∈ [1 − λ , . Note that λ ≥ (1 − λ ) is equivalent to 11 − λ ≤ τ ( K ) = λ − λ − λ .Therefore, by Corollary 1.4, f ( K ) · f ( K ) = [(1 − λ ) , . Since λ ≥ (1 − λ ) , it follows that K · K = ∪ ∞ i =0 λ i ( f ( K ) · f ( K )) ∪ { } = [0 , . Now we prove (3) ⇒ (1). Note that K ÷ K = + ∞ [ n = −∞ λ n f ( K ) f ( K ) ∪ { } ⊂ + ∞ [ n = −∞ λ n (cid:18)(cid:20) − λ , − λ (cid:21)(cid:19) ∪ { } . Therefore, if λ < (1 − λ ) , then (cid:20) − λ , − λ (cid:21) \ (cid:20) λ (1 − λ ) , λ − λ (cid:21) = ∅ . In other words, K ÷ K = R . Finally, we prove (1) ⇒ (3). This step is almost the same as (2) ⇒ (1) interms of the equation K ÷ K = + ∞ [ n = −∞ λ n f ( K ) f ( K ) ∪ { } . Some identities
In this section, we mainly prove Corollary 1.12. It is easy to calculate τ ( F t (7)) = (42 + 24 √ / , t ∈ Z , see [1, Lemma 4.3, Lemma 4.4]. Therefore, by Corollary 1.4, it follows that F (7) + F (7) = " ( 7 + √ + 7 + √ , ( − √
772 ) + − √ . Moreover, it is easy to check that( F (7) + F i (7)) ∩ ( F (7) + F i +1 (7)) = ∅ , i ∈ Z . Therefore, F (7) + F (7) = R . Similarly, we can prove F (7) − F (7) = R . For the second identity, we first note that12 ( C + 1) + F (6) = R ⇔ ( C + 1) + 2 F (6) = R , where ( C + 1) + 2 F (6) = { x + 2 y : x ∈ C + 1 , y ∈ F (6) } . Hence, we only need to prove12 ( C + 1) + F (6) = R . Let f ( x, y ) = 12 x + y, x ∈ C + 1 , y ∈ F (6) . NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 15
Then by Corollary 1.4, we have12 ( C + 1) + F (6)is an interval and (cid:18)
12 ( C + 1) + F i (6) (cid:19) ∩ (cid:18)
12 ( C + 1) + F i +1 (6) (cid:19) = ∅ , i ∈ Z . As such, 12 ( C + 1) + F (6) = R . Finally, we consider the function f ( x, y, z ) = 0 . x + xy + z, x, y ∈ C + 1 , z ∈ F (6) . By Theorem 1.1, we have that f ( C + 1 , C + 1 , F (6)) = " . − √ , . √ . Moreover, f ( C + 1 , C + 1 , F i (6)) ∩ f ( C + 1 , C + 1 , F i +1 (6)) = ∅ , i ∈ Z . Therefore, we have f ( C + 1 , C + 1 , F (6)) = R . Final remarks and some problems
Although in Theorem 1.1, we give a sufficient condition under which thecontinuous image of f is a closed interval, there are many problems left. Welist some problems as follows.(1) For a given E ⊂ R d , define a continuous function g : R d → R d . Itwould be interesting to consider when g ( E ) contains an interior or g ( E ) is exactly some convex hull.(2) In Theorem 1.1, we do not know whether for two concrete sets, thelower and upper bounds of the ratio of partial derivatives can beimproved.(3) In Theorem 1.1, we only consider the first order partial derivatives.Can we give a similar nonlinear version of Theorem 1.1 using higherorders of partial derivatives.(4) In Corollary 1.11, we find an example such that the resonant maxi-mum for the multiplication and division occurs. It would be interest-ing to find more sets which have this resonant phenomenon. More-over, we may consider the resonant phenomenon for other arithmeticoperation such as sum of squares and sum of cubes. These questionsare motivated by the representations of real numbers from numbertheory. (5) Given two Cantor sets K and K with τ ( K ) τ ( K ) <
1, can we findsome sufficient conditions such that f ( K , K ) is still an interval.(6) Given two Cantor sets K and K , we do not know when f ( K , K )is a union of finitely many closed intervals.(7) The Newhouse’s thickness, in some sense, is rough. As it gives arough relation between gaps and bridges. It is deserved to define afiner thickness. Under the new thickness, we may partially improveTheorem 1.1. Acknowledgements
This work is supported by K.C. Wong Magna Fund in Ningbo Univer-sity. This work is also supported by National Natural Science Foundationof China with No. 11701302, and by Zhejiang Provincial Natural ScienceFoundation of China with No.LY20A010009.
References [1] Steve Astels. Cantor sets and numbers with restricted partial quotients.
Trans. Amer.Math. Soc. , 352(1):133–170, 2000.[2] Jayadev S. Athreya, Bruce Reznick, and Jeremy T. Tyson. Cantor set arithmetic.
Amer. Math. Monthly , 126(1):4–17, 2019.[3] Taras Banakh, Eliza Jab lo´nska, and Wojciech Jab lo´nski. The continuity of additiveand convex functions which are upper bounded on non-flat continua in R n . Topol.Methods Nonlinear Anal. , 54(1):247–256, 2019.[4] Karma Dajani and Cor Kraaikamp.
Ergodic theory of numbers , volume 29 of
CarusMathematical Monographs . Mathematical Association of America, Washington, DC,2002.[5] Bohuslav Diviˇs. On the sums of continued fractions.
Acta Arith. , 22:157–173, 1973.[6] Jonathan M. Fraser, Douglas C. Howroyd and Han Yu. Dimension growth for iteratedsumsets.
Math. Z. , 293(3-4):1015–1042, 2019.[7] Dejun Feng and Yufeng Wu. On arithmetic sums of fractal sets in R d . arXiv:2006.12058 , 2020.[8] Jiangwen Gu, Kan Jiang, Lifeng Xi, and Bing Zhao. Multiplication on uniform λ -Cantor set. arXiv:1910.08303 , 2019.[9] Marshall Hall, Jr. On the sum and product of continued fractions. Ann. of Math. (2) ,48:966–993, 1947.[10] James L. Hlavka. Results on sums of continued fractions.
Trans. Amer. Math. Soc. ,211:123–134, 1975.[11] Carlos Gustavo T. de A. Moreira and Jean-Christophe Yoccoz. Stable intersections ofregular Cantor sets with large Hausdorff dimensions.
Ann. of Math. (2) , 154(1):45–96,2001.[12] Sheldon E. Newhouse. The abundance of wild hyperbolic sets and nonsmooth stablesets for diffeomorphisms.
Inst. Hautes ´Etudes Sci. Publ. Math. , (50):101–151, 1979.[13] Jacob Palis and Floris Takens.
Hyperbolicity and sensitive chaotic dynamics at homo-clinic bifurcations , volume 35 of
Cambridge Studies in Advanced Mathematics . Cam-bridge University Press, Cambridge, 1993. Fractal dimensions and infinitely manyattractors.[14] Mehdi.Pourbarat. On the arithmetic difference of middle cantor sets.
Discrete andContinuous Dynamical Systems. , 38(9):4259-4278, 2018.
NONLINEAR VERSION OF THE NEWHOUSE THICKNESS THEOREM 17 [15] Hugo Steinhaus. Mowa W lasno´s´c Mnogo´sci Cantora.
Wector, 1-3. English translationin: STENIHAUS, H.D.
Nonlinearity , 30(5):2114–2137, 2017.[17] Han Yu. Fractal projections with an application in number theory. arXiv:2004.05924 ,2020.(K. Jiang)
Department of Mathematics, Ningbo University, People’s Repub-lic of China
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