We study the effect of addition on the Hamming weight of a positive integer. Consider the first @math positive integers, and fix an @math among them. We show that if the binary representation of @math consists of @math blocks of zeros and ones, then addition by @math causes a constant fraction of low Hamming weight integers to become high Hamming weight integers. This result has applications in complexity theory to the hardness of computing powering maps using bounded-depth arithmetic circuits over @math . Our result implies that powering by @math composed of many blocks require exponential-size, bounded-depth arithmetic circuits over @math .
Kopparty gave a different condition for when @math has the @math -shifting property: its binary representation consists mostly of a repeating constant-length string that is not all zeros or ones @cite_3 . Note that any integer expressible as @math , where @math , @math is odd, and @math , has binary representation of this form. As a consequence, taking @math -th roots and computing @math -th residue symbols cannot be done with polynomial-size @math circuits. Our main result generalizes Kopparty's condition, as the periodic strings form a small subset of the strings with @math blocks. Beck and Li showed that the @math -th residue map is hard to compute in @math by using the concept of algebraic immunity @cite_5 . It is worth noting that their method does not say anything about the complexity of the @math -th root map in @math . So in this regard, there is something to be gained by analyzing the @math -shifting property condition. A more detailed history of the complexity of arithmetic operations using low-depth circuits can be found in @cite_3 .
@cite_5 @cite_3
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. In this paper we study the problem of finding maximally sized subsets of binary strings (codes) of equal length that are immune to a given number @math of repetitions, in the sense that no two strings in the code can give rise to the same string after @math repetitions. We propose explicit number theoretic constructions of such subsets. In the case of @math repetition, the proposed construction is asymptotically optimal. For @math , the proposed construction is within a constant factor of the best known upper bound on the cardinality of a set of strings immune to @math repetitions. Inspired by these constructions, we then develop a prefixing method for correcting any prescribed number @math of repetition errors in an arbitrary binary linear block code. The proposed method constructs for each string in the given code a carefully chosen prefix such that the resulting strings are all of the same length and such that despite up to any @math repetitions in the concatenation of the prefix and the codeword, the original codeword can be recovered. In this construction, the prefix length is made to scale logarithmically with the length of strings in the original code. As a result, the guaranteed immunity to repetition errors is achieved while the added redundancy is asymptotically negligible. Given a discrete random walk on a finite graph @math , the vacant set and vacant net are, respectively, the sets of vertices and edges which remain unvisited by the walk at a given step @math . These sets induce subgraphs of the underlying graph. Let @math be the subgraph of @math induced by the vacant set of the walk at step @math . Similarly, let @math be the subgraph of @math induced by the edges of the vacant net. For random @math -regular graphs @math , it was previously established that for a simple random walk, the graph @math of the vacant set undergoes a phase transition in the sense of the phase transition on Erd os -Renyi graphs @math . Thus, for @math there is an explicit value @math of the walk, such that for @math , @math has a unique giant component, plus components of size @math , whereas for @math all the components of @math are of size @math . We establish the threshold value @math for a phase transition in the graph @math of the vacant net of a simple random walk on a random @math -regular graph. We obtain the corresponding threshold results for the vacant set and vacant net of two modified random walks. These are a non-backtracking random walk, and, for @math even, a random walk which chooses unvisited edges whenever available. This allows a direct comparison of thresholds between simple and modified walks on random @math -regular graphs. The main findings are the following: As @math increases the threshold for the vacant set converges to @math in all three walks. For the vacant net, the threshold converges to @math for both the simple random walk and non-backtracking random walk. When @math is even, the threshold for the vacant net of the unvisited edge process converges to @math , which is also the vertex cover time of the process. We study the complexity of computing the kth-power of an element of F2n by constant depth arithmetic circuits over F2 (also known as ACP). Our study encompasses the complexity of basic arithmetic operations such as computing cube-root and computing cubic-residuosity of elements of F2n. Our main result is that these problems require exponential size circuits. We also derive strong average-case versions of these results. For example, we show that no subexponential-size, constant-depth, arithmetic circuit over F2 can correctly compute the cubic residue symbol for more than 1 3 + o(1) fraction of the elements of F2n. As a corollary, we deduce a character sum bound showing that the cubic residue character over F2n is uncorrelated with all degree-d n-variate F2 polynomials (viewed as functions over F2n in a natural way), provided d l ne for some universal e > 0. Classical methods (based on van der Corput differencing and the Weil bounds) show this only for d l log(n). Our proof revisits the classical Razborov-Smolensky method for circuit lower bounds, and executes an analogue of it in the land of univariate polynomials over F2n. The tools we use come from both F2n and F2n. In recent years, this interplay between F2n and F2n has played an important role in many results in pseudorandomness, property testing and coding theory.