Level sets of the Takagi function: local level sets

The Takagi function ??: [0,1] ?? [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y)?=?{x : ??(x)?=?y} of the Takagi function ??(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a ??generic?? full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a ??generic?? full Lebesgue measure set of abscissas x, the level set L(??(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly ${\frac{3}{2}}$ .     

Weak tangent and level sets of Takagi functions

In this paper, we study some properties of Takagi functions and their level sets. We show that for Takagi functions $$T_{a,b}$$ with parameters a, b such that ab is a root of a Littlewood polynomial, there exist large level sets. As a consequence, we show that for some parameters a, b, the Assouad dimension of graphs of $$T_{a,b}$$ is strictly larger than their upper box dimension. In particular, we can find weak tangents of those graphs with large Hausdorff dimension, larger than the upper box dimension of the graphs.     

Level sets of the Takagi function: Hausdorff dimension

The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity of τ(x) up to affine shifts.     

Level sets of the Takagi function: Hausdorff dimension

The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity of τ(x) up to affine shifts.     

Takagi functions and approximate midconvexity

Let V be a convex subset of a normed space and let ε?0, p>0 be given constants. A function f:V→R is called (ε,p)-midconvex if

     

Approximate convexity of Takagi type functions

Given a bounded function Φ:R→R, we define the Takagi type function TΦ:R→R by

     

Multiple cross-intersecting families of signed sets

A k-signed r-set on[n]={1,…,n} is an ordered pair (A,f), where A is an r-subset of [n] and f is a function from A to [k]. Families A₁,…,Ap are said to be cross-intersecting if any set in any family Ai intersects any set in any other family Aj. Hilton proved a sharp bound for the sum of sizes of cross-intersecting families of r-subsets of [n]. Our aim is to generalise Hilton's bound to one for families of k-signed r-sets on [n]. The main tool developed is an extension of Katona's cyclic permutation argument.     

Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph.     

Distribution of the maxima of random Takagi functions

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ _n=1^∞ ε _n ϕ(2 ⁿ⁻¹ x), x ∈ R, where ɛ ₁, ɛ ₂, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ ₁ = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ _n }, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa ^{n − 1} ɛ _n ϕ(2 ^{n − 1} x), where 0 < a < 1.      

Level sets of differentiable functions of two variables with non-vanishing gradient

We show that if the gradient of exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.     

Critical sets of functions

Summary With the introduction of an alternate definition for critical point, this paper studies critical sets, as defined byW. M. Whyburn, in terms of certain related domains. Critical sets are divided into four classes. Type0 has the limit point property with respect to critical sets which are not type0; type1 and2 critical sets compare, respectively, to classical minimum and maximum points; type3 includes themin-max and flex type. This paper is a result of a study of critical sets made as a dissertation problem under the direction ofW. M. Whyburn, to whom the author is indebted for many helpful suggestions.     

No-where averagely differentiable functions: Baire category and the Takagi function

A classical and well-known result due to Banach and Mazurkiewicz says that a typical (in the sense of Baire) continuous function on the unit interval is no-where differentiable. In this paper we prove that a typical (in the sense of Baire) continuous function $f$ on [0, 1] is spectacularly more irregular than suggested by Banach and Mazurkiewicz’s result. Namely, not only is the difference quotient $\frac{f(x+h)-f(x)}{h}$ divergent as $h\rightarrow 0$ for all $x$ , but the function $h\rightarrow \frac{f(x+h)-f(x)}{h}$ diverges so badly as $h\rightarrow 0$ , that it remains divergent even after being “smoothened out” using iterated Cesaro averages of arbitrary high order. More precisely, we introduce the notion of higher order average differentiability based on iterated Cesaro averages and prove that not only is a typical (in the sense of Baire) continuous function on [0, 1] no-where differentiable, but it is even no-where averagely differentiable of any order. We also show that the no-where differentiable Takagi function is, in fact, no-where averagely differentiable.     

Dimension functions of Cantor sets

We estimate the packing measure of Cantor sets associated to non-increasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions recently found by Cabrelli et al for these sets.