RESEARCH STATEMENT: MUSHFEQ KHAN

RESEARCH STATEMENT: MUSHFEQ KHAN Contents 1. Overview 1 2. Notation and basic definitions 4 3. Shift-complex sequences 4 4. PA degrees and slow-growing DNC functions 5 5. Lebesgue density and Π 0 1 classes 8 References 9 My specialty is mathematical logic, specifically computability theory, which charts the variety of ways in which mathematical objects can be impossible to compute, or difficult to describe, or possibly useful for computing other mathematical objects that are impossible to compute outright. My work mainly inhabits the areas of algorithmic randomness and classical computability theory, although I have also worked in reverse mathematics. Algorithmic randomness is concerned with different formulations of what it means for an infinite binary sequence to be effectively random, or for one such sequence to be more random than another. The tools and concepts in algorithmic randomness are diverse, but they tend to be analytic in character. On the other hand, classical computability theory is concerned with questions about the structure of the Turing degrees and about notions of complexity that are generally discrete and combinatorial in their flavor. Some extremely interesting problems emerge when one considers how these two apparently different styles of complexity interact, and these have been the primary focus of my research. 1. Overview From relatively obscure origins in the work of Osvald Demuth, Per Martin-Löf, and C.P. Schnorr, algorithmic randomness has exploded into one of the most active areas in computability theory. In the process, it has introduced a wealth of new ideas and techniques into the core of the field, some of which come from other parts of mathematics: for example, from information theory, the concept of Kolmogorov complexity; from geometric measure theory, the notion of Hausdorff dimension; and from probability theory, the notion of martingale. Kolmogorov complexity, which we denote by K, is a measure of the information content or randomness of a finite binary string. We fix a suitable universal machine U that converts strings to strings (somewhat like an idealized Unzip program). Then for a string σ, K(σ) is the least length of an input to U that yields σ as the output. Martin-Löf originally defined his formulation of the randomness of an infinite binary sequence in terms of tests that generalize the idea of a statistical test for randomness. In this paradigm, a statistical regularity property, i.e., one which a random sequence should avoid, is a property that can be isolated in Cantor space by an effectively represented measure 0 set. It is a striking theorem of Schnorr s that Kolmogorov complexity can capture exactly the same idea: For an infinite sequence X, let X n denote the initial segment of X of length n. 1

2 RESEARCH STATEMENT: MUSHFEQ KHAN Then a sequence X is Martin-Löf random if and only if for some c N, for every length n, K(X n) n c. 1.1. Shift-complexity. One of the properties we would expect of any random sequence is normality: for each n, all binary strings of length n should occur as substrings of the sequence with equal asymptotic frequency. Any such sequence would necessarily contain substrings that have extremely low Kolmogorov complexity relative to their lengths. For example, for any k N, a string of 10 k consecutive 0s must occur at some point (in fact, infinitely often). It is not hard to see that the Kolmogorov complexity of such a string should be on the order of log 2 (k) or less. Shift-complex sequences are those which avoid this deficiency: they are fairly complex no matter where we look. Formally, if X is shift-complex, then there is a uniform coefficient δ (0, 1) and a c N such for any substring σ of X, K(σ) δ σ c. A shift-complex sequence with coefficient δ is called δ-shift-complex. It is a consequence of Schnorr s theorem that we cannot allow δ to be 1 in this definition. It is not at all obvious that shift-complex sequences even exist: building such a sequence constitutes a one-dimensional tiling problem where one must be careful when placing the tiles not to introduce forbidden intermediate strings, specifically those of low complexity. Two constructions, including one due to Leonid Levin, are presented in my survey on the subject [13]. The study of shift-complexity is very new, but we already know a few key facts about these sequences. The considerations mentioned above show that the class of shift-complex sequences and the class of Martin-Löf random sequences are disjoint. But more is true: shift-complexity and Martin-Löf randomness are, in a precise sense, orthogonal properties. We say a sequence X computes a sequence Y (or is Turing above Y, written Y T X) if there is a program that implements the characteristic function of Y given black-box access to the characteristic function of X. Rumyantsev showed that not every Martin-Löf random sequence computes a shift-complex sequence. I showed that the other direction fails as well: Theorem 1.1 (Khan [13]). For every δ (0, 1) there is a δ-shift-complex sequence that computes no Martin-Löf random sequence. Many questions about shift-complexity remain open, and some of these are the subject of Section 3. Further, shift-complexity is closely related to the field of symbolic dynamics. For a given δ (0, 1) and c N, the class of shift-complex sequences with coefficient δ and constant c are an example of a subshift, a kind of discrete dynamical system studied in symbolic dynamics. The computational aspects of more general classes of subshifts have only just begun to be studied by computability theorists, notably Simpson. 1.2. Diagonal noncomputability. The technique that allows us to drive a wedge between shift-complexity and Martin-Löf randomness in Theorem 1.1 is an analysis of the computational power of diagonally noncomputable (DNC) functions of varying growth rates. DNCness is intimately related to shift-complexity and Martin-Löf randomness: All shift-complex sequences and Martin-Löf random sequences compute DNC functions. Let J : N N denote the diagonal partial computable function, which we postpone defining in detail for now. The salient features of J are that it can be implemented by a program, that it is partial (its domain is not all of N), and that it is universal: it interprets all other functions that can be implemented by a program. A DNC function is a total function f from N to N that never agrees with J wherever the latter is defined; f always guesses wrong about J.

RESEARCH STATEMENT: MUSHFEQ KHAN 3 It turns out that by constraining the amount of freedom that f has to guess wrong to a finite set of options, one obtains a very powerful property. Such an f is said to be of PA degree: among other things, it can compute a complete, consistent extension of Peano Arithmetic. The main goal now is to understand the difference in computational power between the general class of DNC functions and the ones of PA degree. Ambos-Spies, et al [1], and Greenberg and Miller [9] have introduced a new approach to classifying DNC functions in terms of how quickly they grow. This approach has already had useful applications, including in Theorem 1.1. One of the main tools involved is bushy tree forcing, a method for building DNC functions pioneered by Kumabe and Lewis [17]. My work [15] analyzes and extends this technique to produce a stronger result (Theorem 4.6), and constitutes a first step towards addressing the questions I consider in Section 4. 1.3. Lebesgue density and Π 0 1 classes. Analyzing the effective content of the Lebesgue density theorem played a crucial role in some recent developments in algorithmic randomness, namely the solutions of the ML-covering and ML-cupping problems [4,7, 8]. Two new classes of sequences emerged from this analysis: the positive density points with respect to effectively closed (or Π 0 1 ) sets of reals, and a proper subclass, the density-one points. Strictly speaking, being density-one is neither a randomness nor a genericity property. However, it has a combinatorial characterization that appears to be a natural generalization of 1-genericity. What sorts of properties of 1-generics remain true of density-one points? Within the Martin- Löf random sequences, the positive density points correspond to a familiar randomness class: the Turing incomplete (or difference random) sequences (Bienvenu, Hölzl, Miller, and Nies [5]). What happens to this equivalence when we remove the assumption of randomness? My work on density answers these questions, and is summarized in Section 5. 1.4. Reverse mathematics of lattice representation theorems. Reverse mathematics concerns itself with calibrating the strength of theorems as axioms. Given a theorem, it asks which other theorems imply it, and which theorems it implies. This would be a meaningless question when working within a system as powerful as ZFC, which proves everything outright. So reverse mathematics adopts a very weak base system, known as RCA 0, which is able to formalize just computable mathematics. Brodhead, Kjos-Hanssen, Lampe, Nguyen, Shore, and I [6] have investigated the reverse mathematical strength of a representation theorem in lattice theory known as the Grätzer- Schmidt theorem. It says that for each algebraic lattice L, there is an algebra A such that L is isomorphic to the lattice of congruences of A. Our work builds on P. Pudlák s short proof [18] of the theorem. While the question of the precise strength of the strongest form of the theorem remains open, we have shown that the version of the theorem restricted to distributive algebraic lattices is implied by the axiom system ACA 0. Key to this result is a study of the complexity of the set of compact elements of a computable distributive algebraic lattice. Using characterizations of compactness that draw on lattice theory, we have shown that this set is Π 0 3 and can be Π 0 3 -complete. 1.5. Effective bi-immunity. An infinite set X of natural numbers is said to be immune if it contains no infinite c.e. subset. It is said to be effectively immune if there exists a computable function f such that for every e, if W e X, then W e < f(e). Jockusch [11] has shown that every DNC function is Turing equivalent to an effectively immune set. More recently, Jockusch and Lewis [12] have shown that every DNC function computes a bi-immune set, i.e. one such that both it and its complement are immune. They asked if

4 RESEARCH STATEMENT: MUSHFEQ KHAN every DNC function computes an effectively bi-immune set, i.e., one such that both it and its complement are effectively immune. This, surprisingly, turned out to not be the case (Beros [2]). With Beros and Kjos-Hanssen [3], I have shown further that there is a real of effective Hausdorff dimension 1 that computes no effectively bi-immune set. This generalizes 1 Greenberg and Miller s result [9] on the existence of reals of effective Hausdorff dimension 1 that compute no Martin-Löf random real. The question of whether every effectively bi-immune set computes a Martin-Löf random real remains open. 1.6. Strongly noncomputable functions. A function f : ω ω is said to be strongly noncomputable (or SNC) if for each partial computable function ψ, f(n) = ψ(n) for only finitely many n. If h is computable function let SNC h denote the class of SNC functions that are bounded by h. In forthcoming work with Beros, Kjos-Hanssen, and Nies, I have studied the Muchnik (or weak) degrees of these classes. We have shown that one obtains a proper hierarchy of these classes as one varies the computable bound h. Many open questions about the Muchnik sublattice comprised of these classes remain. 2. Notation and basic definitions We need some elementary concepts from computability theory. By a sequence we mean an element of 2 N, i.e., an infinite sequence of binary digits indexed by N. Let N N denote the space of functions from N to N. A partial function from N to N is partial computable if there is an algorithm that implements the function. There are countably many algorithms that correspond to a partial computable function, and they can be listed effectively. Let (ϕ n ) n N be such a listing. The diagonal partial computable function J is of particular importance to us: J(n) = ϕ n (n). Note that J itself is partial computable. The domain of J is a well-known set, the halting problem. We denote it by 0. A partial computable function that is total is said to be computable. A sequence is computable if it is computable when viewed as the characteristic function of a set of natural numbers. More generally, we can speak of computable sets of objects that can be coded by natural numbers (e.g., sets of finite strings or rationals). A set is computably enumerable, or c.e., if there is an algorithm that lists its elements in some order. A set of sequences U 2 N is a Σ 0 1 class if there is a c.e. set of finite binary strings W such that U consists exactly of the sequences that have an initial segment in W. The complement of a Σ 0 1 class is a Π0 1 class. The preorder T was introduced above; it induces the equivalence relation T. The equivalence classes of sequences under T are known as the Turing degrees. All definitions involving T carry over in a natural way to functions in N N, since sequences can code functions and vice versa. 3. Shift-complex sequences Rumyantsev showed that the measure of the set of sequences that compute shift-complex sequences is 0. Since the set of Martin-Löf random sequences has full measure, this immediately implies that there are Martin-Löf random sequences that do not compute any 1 Every Martin-Löf random real is effectively bi-immune.

RESEARCH STATEMENT: MUSHFEQ KHAN 5 shift-complex sequence. But there are Martin-Löf random sequences that compute the halting problem, and it is not difficult to see that these do compute shift-complex sequences. These are, in fact, the only ones: Theorem 3.1 (Khan [13]). A Martin-Löf random sequence computes a shift-complex sequence if and only if it computes the halting problem. Together with Theorem 1.1, this establishes how shift-complexity and Martin-Löf randomness relate to each other in the setting of the Turing degrees. However, we do not yet have a complete picture of how shift-complexity relates to another fundamental concept in algorithmic randomness: effective Hausdorff dimension. 3.1. Dimension extraction. The effective Hausdorff dimension of a sequence X, denoted by dim(x), measures its information density. It is an effectivization of the notion of Hausdorff dimension from geometric measure theory, which assigns a fractional dimension to any measure 0 set of reals and calibrates the complexity of the set. A remarkable theorem of Mayordomo connects this measure-theoretic concept to Kolmogorov complexity. For any sequence X, K(X n) dim(x) = lim inf. n n Clearly, for a δ-shift-complex sequence X, dim(x) δ. But to what extent can we extract dimension from such a sequence? Question 3.2. Does every shift-complex sequence compute a sequence of effective Hausdorff dimension 1? If not, then does it compute one of dimension (1 ε) for every ε > 0? 3.2. Complexity extraction. Hirschfeldt and Kach [10] have shown that every δ-shiftcomplex sequence computes a δ -shift-complex sequence for some δ (δ, 1). A key question is to what extent complexity extraction is possible: Question 3.3. Fix δ (0, 1). Does every δ-shift-complex sequence compute a δ -shiftcomplex sequence for every δ (δ, 1)? 3.3. Bi-infinite shift-complex sequences. A bi-infinite shift-complex sequence is a shiftcomplex sequence where the bits are indexed by Z. It is known that a δ-shift-complex sequence computes a bi-infinite one of slightly lower complexity: Proposition 3.4 (Khan [13]). Fix ε (0, 1/2). Every (1 ε)-shift-complex sequence computes a bi-infinite (1 2ε)-shift-complex sequence. Question 3.5. For every δ (0, 1), does every δ-shift-complex sequence compute a bi-infinite one of the same complexity? 4. PA degrees and slow-growing DNC functions A function f N N is called diagonally noncomputable (or DNC), if f(n) J(n) for all n in the domain of J. Such a function can be thought of as witnessing its noncomputability in a very uniform way. DNC functions are of fundamental importance in computability theory. Moreover, they are ubiquitous: almost every sequence (in fact, every Martin-Löf random sequence) computes such a function. Because it is a full measure property on Cantor space, we think of DNC-ness on its own as a very weak form of noncomputability. However, by constraining the values of a DNC function to a finite set (i.e., allowing it a finite amount of freedom to diagonalize against J), one obtains a very strong property. For k N, let DNC k

6 RESEARCH STATEMENT: MUSHFEQ KHAN denote the class of DNC functions that always take values less than k. We say a function f N N is of PA degree if it computes a DNC 2 function. PA degrees are so named because they compute consistent, complete extensions of Peano Arithmetic, but they also have many other interesting properties. For example, they compute an element of every nonempty Π 0 1 class. These are both, in fact, alternative characterizations. A third was introduced in Subsection 1.2: Jockusch [11] presents a proof, due to Friedberg, of the surprising fact that for each k > 2 any DNC k function computes a DNC 2 function, and so all functions that diagonalize against J using a bounded amount of freedom are of PA degree. However, this is no longer the case when we allow a DNC function to grow unboundedly. For our purposes, an order function is a computable function from N to N \ {0, 1} that is nondecreasing and unbounded. For an order function h, let DNC h denote the class of DNC functions that are bounded above pointwise by h. The heuristic to keep in mind is that the slower the growth of h, the more powerful the functions in DNC h are. If a computability-theoretic property P is shared by all members of DNC h for some order function h, then we say that sufficiently slow-growing DNC functions have property P. It is a straightforward application of bushy tree forcing to see that this is not true where P is the property of having PA degree: Given any order function h, there is a function in DNC h that is not of PA degree. The broad question, then, is the following: Question 4.1. What sorts of properties of PA degrees are true of sufficiently slow-growing DNC functions? What makes Question 4.1 compelling is that we already have some surprising answers, but as yet, little insight into the general phenomenon. Let us first consider a few examples. 4.1. Martin-Löf random and Kurtz random sequences. There is a Π 0 1 class consisting solely of Martin-Löf random sequences. Therefore, every PA degree computes a Martin- Löf random sequence. Greenberg and Miller [9] showed that this is not true of sufficiently slow-growing DNC functions. Kurtz randomness 2 is a much weaker property than Martin-Löf randomness. Using bushy tree forcing, Miller and I proved the following strengthening of the Greenberg-Miller theorem: Theorem 4.2 (Khan, Miller [15]). For every order function h, there is a DNC h function that computes no Kurtz random sequence. 4.2. Effective Hausdorff dimension 1 and shift-complex sequences. Greenberg and Miller [9] showed that sufficiently slow-growing DNC functions compute sequences of effective Hausdorff dimension 1. I showed: Theorem 4.3 (Khan [13]). For every δ (0, 1), there is an order function h such that all DNC h functions compute a δ-shift-complex sequence. 4.3. Minimal degrees. We say a sequence X is minimal if it is noncomputable and for any sequence Y T X, either Y is computable or X T Y. Kumabe and Lewis have [17] constructed a minimal DNC function. However, this DNC function grows very quickly, leaving the following question open: Question 4.4. Are there minimal DNC h functions for every order function h? 2 A sequence is Kurtz random if it is not contained in any measure 0 Π 0 1 class.

RESEARCH STATEMENT: MUSHFEQ KHAN 7 Constructing such a function would require a significant refinement of the techniques in [17]. Question 4.4 is interesting not just because it is an instance of Question 4.1, but also because it is related to the following open problem: Question 4.5. What is the classical Hausdorff dimension of the set of minimal sequences? As mentioned above, Greenberg and Miller have exhibited an order function h such that every DNC h function computes a sequence of effective Hausdorff dimension 1. If Question 4.4 were to have a positive answer (or at least, a positive answer for the order function h ), then it would mean that there exists a minimal sequence of effective Hausdorff dimension 1. However, in order to address Question 4.5 (which asks about classical Hausdorff dimension), we would have to prove an even stronger statement: that for any sequence X, there is a h -bounded function that is DNC relative to X (we denote such functions by DNC X h ) and is minimal. With this goal in mind, I proved a strengthening of the Kumabe-Lewis theorem: Theorem 4.6 (Khan [15]). There is an order function g such that for every oracle X, there is a minimal DNC X g function. For this proof to work, we need g to grow quite fast, so the Greenberg-Miller theorem does not apply. Closing the gap between g and h remains a problem I am deeply interested in. 4.4. The cupping property and a problem of Yates. If X and Z are sequences then let X Z denote the sequence formed by alternating the bits of X and Z. We write X > T Y if X computes Y but not vice versa. A sequence X has the cupping property if for every Y such that Y > T X, there is a Z < T Y such that X Z T Y. The cupping property should be thought of as a form of computational strength: X possesses some nontrivial information about every sequence that is properly Turing above it. Kučera [16] showed that every sequence of PA degree has the cupping property. This yields another instance of Question 4.1: Question 4.7. Do sufficiently slow-growing DNC functions have the cupping property? The cupping property is of fundamental interest to the study of the structure of the Turing degrees. One of the oldest questions in this area, due to Yates, is whether every minimal sequence has a strong minimal cover. The cupping property and the property of having a strong minimal cover are mutually exclusive. Thus, if the answer to Question 4.7 turns out to be yes, then we would have a new line of attack on Yates s problem. 4.5. Computing members of Π 0 1 classes. Many properties are true of PA degrees by virtue of the fact that they are true of all members of some Π 0 1 class. A more general approach to answering Question 4.1 might then be to investigate which Π 0 1 classes contain members that are computable from sufficiently slow-growing DNC functions: Question 4.8. Given an order function h, can we characterize the Π 0 1 classes C such that every DNC h function computes an element of C? 4.6. The DNC degrees. Ambos-Spies, et al [1] were the first to recognize a hierarchy of DNC strength : They showed that for every order function g there is a faster growing order function h, and a DNC h function that computes no DNC g function. With Cai, I showed: Theorem 4.9 (Cai, Khan). Given any order function g 0, there is another order function g 1 and functions f 0 DNC g0 and f 1 DNC g1 such that f 0 computes no DNC g1 function and f 1 computes no DNC g0 function.

8 RESEARCH STATEMENT: MUSHFEQ KHAN This suggests the following preordering on the class of order functions: we say h DNC g if every DNC g function computes a DNC h function. The preordering induces the equivalence relation DNC, and we refer informally to the resulting equivalence classes as the DNC degrees. Theorem 4.9 then says simply that given any DNC degree, we can find an incomparable one. We would like to better understand the structure of the DNC degrees. Specific questions include whether they are dense and whether they contain maximal elements. 5. Lebesgue density and Π 0 1 classes The Lebesgue density theorem says that if A is any Lebesgue measurable set of reals, for almost every point x of A, the density of A at x is 1. Informally, the more we zoom in on x by looking at a smaller and smaller interval containing it, the closer to 1 is the fractional measure of A within that interval. Suppose that C is a countable collection of Lebesgue measurable subsets of the unit interval. We say x [0, 1] is a density-one point for C if for every P C that contains x, the density of P at x is 1. It follows from the Lebesgue density theorem (and the countable additivity of Lebesgue measure) that almost every point in [0, 1] is a density-one point for C. Of particular interest are the density-one points we obtain when C is the collection of effectively closed (or Π 0 1 ) subsets of [0, 1]. These have been at the heart of several recent developments in algorithmic randomness, such as the solutions of the ML-covering and ML-cupping problems [7, 8]. An interesting fact that emerged from this line of research is that the Martin-Löf random density-one points of Π 0 1 classes are computationally weak: they do not compute the halting problem. I showed (Theorem 5.3) that this is no longer true when the assumption of Martin-Löf randomness is removed. 1-generics, which are one of the most widely studied class of sequences in computability theory, are closely connected to the density-one points of Π 0 1 classes. In fact, every 1-generic is a density-one point: If a 1-generic is a member of a Π 0 1 class P, then there is an open interval around it that is contained in P. A general density-one point can then be viewed as a more tolerant 1-generic: it permits P to have gaps in the interval, as long as the gaps are not too big in fractional measure, and this measure goes down as we shrink the interval. 5.1. Dyadic density vs full density. There are at least two different ways in use of measuring the density of a set C at a real x: dyadic density and full density. The distinction hinges on what sorts of intervals we consider when we zoom in around x. In Cantor space, dyadic intervals, of the form [n 2 k, (n + 1) 2 k ], seem like the natural choice, even though these are very restrictive when compared to the arbitrary intervals we consider when working on the unit interval. Let µ denote the uniform measure on Cantor space as well as Lebesgue measure on the unit interval (they are measure-theoretically isomorphic). If σ is a finite binary string, let [σ] denote the set of sequences in 2 N that have σ as an initial segment. Then for a sequence X and a measurable set C 2 N, the dyadic density of C at X, written ρ 2 (C X), is lim inf n µ(c [X n]). µ([x n]) We say X is a dyadic positive density point if for every Π 0 1 class C that contains X, ρ 2(C X) > 0. We say X is a dyadic density-one point if for every Π 0 1 class C that contains X, ρ 2(C X) = 1.

RESEARCH STATEMENT: MUSHFEQ KHAN 9 For a real number x [0, 1] and a measurable set C [0, 1], the full density of C at x, written ρ(c x), is µ((x γ, x + δ) C) lim inf. γ,δ 0 + γ + δ On the unit interval, we can define Σ 0 1 classes in a manner analogous to how they are defined in Cantor space: they are the union of a c.e. set of open intervals with rational endpoints. The complement of such a Σ 0 1 class is what we mean by a Π0 1 class on the unit inverval. We say x is a full positive density point if for every Π 0 1 class C [0, 1] that contains x, ρ(c x) > 0. We say x is a full density-one point if for every Π 0 1 class C [0, 1] that contains x, ρ(c x) = 1. If x is irrational, we can uniquely identify it with a binary sequence, namely its binary expansion. So it makes sense to ask if x is a dyadic density-one point. Likewise, it makes sense to ask if a sequence X 2 N is a full density-one point. It is not hard to see that every full density-one point is a dyadic density-one point, but the converse is not true: Theorem 5.1 (Khan [14]). There is a dyadic density-one point that is not full positive density. However, the two notions are equivalent on the class of Martin-Löf random sequences: Theorem 5.2 (Khan, Miller [14]). If X is Martin-Löf random and a dyadic density-one point, then it is a full density-one point. 5.2. Computational strength. Bienvenu, Hölzl, Miller, and Nies [5] showed that the Martin- Löf random dyadic positive density points cannot compute the halting problem. One might ask is if this holds in the absence of randomness. A natural example of a nonrandom densityone point is a 1-generic, but 1-generics cannot compute the halting problem either. It is, however, possible to build a density-one point that is Turing above any sequence: Theorem 5.3 (Khan [14]). For every X 2 N, there is a full density-one point Y such that X T Y T X 0. 5.3. Computing 1-generics. Since 1-generics and density-one points are closely related, it is natural to try to determine what properties they have in common. For example, 1-generics are easily seen to be non-minimal because if X Y is 1-generic, then X and Y are Turing incomparable. This fails for density-one points: Proposition 5.4 (Khan [14]). There is a dyadic density-one point X Y with X T Y. Can a dyadic positive density point be of minimal Turing degree? I recently answered this question in the negative: Theorem 5.5 (Khan [14]). Every dyadic positive density point is either Martin-Löf random or computes a 1-generic. References [1] Klaus Ambos-Spies, Bjørn Kjos-Hanssen, Steffen Lempp, and Theodore A. Slaman. Comparing DNR and WWKL. J. Symbolic Logic, 69(4):1089 1104, 2004. [2] A. A. Beros. A DNC function that computes no effectively bi-immune set. ArXiv e-prints, August 2013. [3] Achilles Beros, Mushfeq Khan, and Bjørn Kjos-Hanssen. Effective bi-immunity and randomness. In Downey Festschrift, volume 10010 of Lecture Notes in Comput. Sci. Springer, Berlin, 2016. To appear. [4] Laurent Bienvenu, Adam R. Day, Noam Greenberg, Antonín Kučera, Joseph S. Miller, André Nies, and Dan Turetsky. Computing K-trivial sets by incomplete random sets. Bull. Symb. Log., 20(1):80 90, 2014. [5] Laurent Bienvenu, Rupert Hölzl, Joseph S. Miller, and André Nies. Denjoy, Demuth and density. J. Math. Log., 14(1):1450004, 35, 2014.

10 RESEARCH STATEMENT: MUSHFEQ KHAN [6] Katie Brodhead, Mushfeq Khan, Bjørn Kjos-Hanssen, William A. Lampe, Paul Kim Long V. Nguyen, and Richard A. Shore. The strength of the Grätzer-Schmidt theorem. Arch. Math. Logic, 55(5-6):687 704, 2016. [7] Adam R. Day and Joseph S. Miller. Cupping with random sets. Proc. Amer. Math. Soc., 142(8):2871 2879, 2014. [8] Adam R. Day and Joseph S. Miller. Density, forcing, and the covering problem. Math. Res. Lett., 22(3):719 727, 2015. [9] Noam Greenberg and Joseph S. Miller. Diagonally non-recursive functions and effective Hausdorff dimension. Bull. Lond. Math. Soc., 43(4):636 654, 2011. [10] Denis Hirschfeldt and Asher Kach. Shift complex sequences. Talk presented at the AMS Eastern Sectional Meeting, 2012. [11] Carl G. Jockusch, Jr. Degrees of functions with no fixed points. In Logic, methodology and philosophy of science, VIII (Moscow, 1987), volume 126 of Stud. Logic Found. Math., pages 191 201. North-Holland, Amsterdam, 1989. [12] Carl G. Jockusch, Jr. and Andrew E. M. Lewis. Diagonally non-computable functions and bi-immunity. J. Symbolic Logic, 78(3):977 988, 2013. [13] Mushfeq Khan. Shift-complex sequences. Bull. Symbolic Logic, 19(2):199 215, 2013. [14] Mushfeq Khan. Lebesgue density and 0 1 classes. J. Symb. Log., 81(1):80 95, 2016. [15] Mushfeq Khan and Joseph S. Miller. Forcing with bushy trees. Submitted. [16] Antonín Kučera. Measure, Π 0 1-classes and complete extensions of PA. In Recursion theory week (Oberwolfach, 1984), volume 1141 of Lecture Notes in Math., pages 245 259. Springer, Berlin, 1985. [17] Masahiro Kumabe and Andrew E. M. Lewis. A fixed-point-free minimal degree. J. Lond. Math. Soc. (2), 80(3):785 797, 2009. [18] Pavel Pudlák. A new proof of the congruence lattice representation theorem. Algebra Universalis, 6(3):269 275, 1976.