Sumset Research Articles

AN EXPLORATION OF ANTIMULTIGROUP EXTENSIONS

This research paper pioneers an innovative extensions of antimultigroup theory by seemingly integrating the concept of cuts and comultiset, thereby revolutionizing the field. Notably, we demonstrate that the root sets of antimultigroup sums and differences are subgroups, uncovering a profound connection. Furthermore, we establish that if is a complete sub-antimultigroup of such that all the counts in are factors of their corresponding counts in . Finally, we prove that the cuts of antimultigroup unions and intersections also form subgroups, further enriching our understanding of these complex structures.

A linear time approach to three-dimensional reconstruction by discrete tomography

The goal of discrete tomography is to reconstruct an unknown function $f$ via a given set of line sums. In addition to requiring accurate reconstructions, it is favourable to be able to perform the task in a timely manner. This is complicated by the presence of ghosts, which allow many solutions to exist in general. In this paper we consider the case of a function $f : A \to \mathbb{R}$ where $A$ is a finite grid in $\mathbb{Z}^3$. Previous work has shown that in the two-dimensional case it is possible to determine all solutions in parameterized form in linear time (with respect to the number of directions and the grid size) regardless of whether the solution is unique. In this work, we show that a similar linear method exists in three dimensions under the condition of nonproportionality. We show that the condition of nonproportionality is fulfilled in the case of three-dimensional boundary ghosts.

A note on the exceptional set for sums of unlike powers of primes

A note on the exceptional set for sums of unlike powers of primes

Weighted sums and Berry-Esseen type estimates in free probability theory

We study weighted sums of free identically distributed self-adjoint random variables with weights chosen randomly from the unit sphere and show that the Kolmogorov distance between the distribution of such a weighted sum and Wigner’s semicircle law is of order n-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-\\frac{1}{2}}$$\\end{document} with high probability. Replacing the Kolmogorov distance by a weaker pseudometric, we obtain a rate of convergence of order n-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n^{-1}$$\\end{document}, thus providing a free analog of the Klartag-Sodin result in classical probability theory. Moreover, we show that our ideas generalize to the setting of sums of free non-identically distributed bounded self-adjoint random variables leading to a new rate of convergence in the free central limit theorem.

On metric approximate subgroups

Let [Formula: see text] be a group with a metric invariant under left and right translations, and let [Formula: see text] be the ball of radius [Formula: see text] around the identity. A [Formula: see text]-metric approximate subgroup is a symmetric subset [Formula: see text] of [Formula: see text] such that the pairwise product set [Formula: see text] is covered by at most [Formula: see text] translates of [Formula: see text]. This notion was introduced in [T. Tao, Product set estimates for noncommutative groups, Combinatorica, 28(5) (2008) 547–594, doi:10.1007/s00493-008-2271-7; T. Tao, Metric entropy analogues of sum set theory (2014), https://terrytao.wordpress.com/2014/03/19/metric-entropy-analogues-of-sum-set-theory/] along with the version for discrete groups (approximate subgroups). In [E. Hrushovski, Stable group theory and approximate subgroups, J. Amer. Math. Soc. 25(1) (2012) 189–243, doi:10.1090/S0894-0347-2011-00708-X], it was shown for the discrete case that, at the asymptotic limit of [Formula: see text] finite but large, the “approximateness” (or need for more than one translate) can be attributed to a canonically associated Lie group. Here we prove an analogous result in the metric setting, under a certain finite covering assumption on [Formula: see text] replacing finiteness. In particular, if [Formula: see text] has bounded exponent, we show that any [Formula: see text]-metric approximate subgroup is close to a [Formula: see text]-metric approximate subgroup for an appropriate [Formula: see text].

Two problems on subset sums

For a set A of positive integers, let P(A) denote the set of all finite subset sums of A. In this paper, we completely solve a problem of Chen and Wu by proving that if B={b1<b2<⋯} is a sequence of integers with b1≥11, 3b1+5≤b2≤4b1, 3b2+2≤b3≤3b2+b1 and 3bn−bn−2≤bn+1≤3bn(n≥3), then there exists a set of positive integers A for which P(A)=N∖B. We also partially answer a problem of Wu by determining the structure of B={b1<b2<⋯} with b1>10 and b2>3b1+4, for which there exists a set of positive integers A such that P(A∩[0,bk])=[0,2bk]∖{bi,2bk−bi:1≤i≤k}(k≥2).

Some properties of classes of minimal numberings of arithmetical set families

We prove that for each u ⩾ 2 the class of all single-valued Σ0 ucomputable numberings of any infinite family of total functions is effectively infinite and the class of all its Σ0 u-1-computable numberings is generated by the downward closure with respect to the reducibility of the set of all infinite direct sums of uniformly Σ0 u-1-computable sequences of its single-valued numberings. It is established that if u &gt; 2, then the class of all Σ0 u-computable numberings of any infinite family is generated by infinite direct sums of uniformly Σ0 u-computable and uniformly Σ0 u-minimal sequences of its numberings.

A conjecture of Hegyvári

For a given sequence [Formula: see text] of nonnegative integers, let [Formula: see text] be the set of all finite subsequence sums of [Formula: see text]. [Formula: see text] is called complete if [Formula: see text] contains all sufficiently large integers. A real number [Formula: see text] is called as an infinite diadical fraction (briefly i.d.f.) if the digit 1 appears infinitely many times in the binary representation of [Formula: see text]. Hegyvári conjectured that [Formula: see text] is complete if [Formula: see text] or [Formula: see text] is i.d.f. and [Formula: see text], where [Formula: see text] is a sequence of integers. In this paper, we give a partial result of Hegyvári’s conjecture.

Convexity, Squeezing, and the Elekes-Szabó Theorem

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szabó Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that\[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\]We are also able to prove that\[\max \{|A+ A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\]Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.

Several classes of new projective three-weight or four-weight linear codes and their applications in $ s $-sum sets

In this paper, let $ q $ be a power of a prime, we construct several classes of new projective three-weight or four-weight linear codes over $ \mathbb{F}_q $ from the defining sets construction, and determine their weight distributions by using additive character sums. Especially, these codes are suitable for applications in secret sharing schemes. Furthermore, basing on two classes of projective three-weight codes, we construct some $ s $-sum sets for any odd $ s&gt;1 $.

Hardness and approximation of submodular minimum linear ordering problems

AbstractThe minimum linear ordering problem (MLOP) generalizes well-known combinatorial optimization problems such as minimum linear arrangement and minimum sum set cover. MLOP seeks to minimize an aggregated cost $$f(\cdot )$$ f ( · ) due to an ordering $$\sigma $$ σ of the items (say [n]), i.e., $$\min _{\sigma } \sum _{i\in [n]} f(E_{i,\sigma })$$ min σ ∑ i ∈ [ n ] f ( E i , σ ) , where $$E_{i,\sigma }$$ E i , σ is the set of items mapped by $$\sigma $$ σ to indices [i]. Despite an extensive literature on MLOP variants and approximations for these, it was unclear whether the graphic matroid MLOP was NP-hard. We settle this question through non-trivial reductions from mininimum latency vertex cover and minimum sum vertex cover problems. We further propose a new combinatorial algorithm for approximating monotone submodular MLOP, using the theory of principal partitions. This is in contrast to the rounding algorithm by Iwata et al. (in: APPROX, 2012), using Lovász extension of submodular functions. We show a $$(2-\frac{1+\ell _{f}}{1+|E|})$$ ( 2 - 1 + ℓ f 1 + | E | ) -approximation for monotone submodular MLOP where $$\ell _{f}=\frac{f(E)}{\max _{x\in E}f(\{x\})}$$ ℓ f = f ( E ) max x ∈ E f ( { x } ) satisfies $$1 \le \ell _f \le |E|$$ 1 ≤ ℓ f ≤ | E | . Our theory provides new approximation bounds for special cases of the problem, in particular a $$(2-\frac{1+r(E)}{1+|E|})$$ ( 2 - 1 + r ( E ) 1 + | E | ) -approximation for the matroid MLOP, where $$f = r$$ f = r is the rank function of a matroid. We further show that minimum latency vertex cover is $$\frac{4}{3}$$ 4 3 -approximable, by which we also lower bound the integrality gap of its natural LP relaxation, which might be of independent interest.

On arithmetic sums of Cantor-type sequences of integers

We are looking for integer sets that resemble classical Cantor set and investigate the structure of their sum sets. Especially we investigate FS(B) the subset sum of sequence type B={⌊pnα⌋}n=0∞. When p=2, then we prove FS(B)+FS(B)=N by analogy with the Cantor set, and some structure theorem for p>2.

Chowla and Sarnak conjectures for Kloosterman sums

AbstractWe formulate several analogs of the Chowla and Sarnak conjectures, which are widely known in the setting of the Möbius function, in the setting of Kloosterman sums. We then show that for Kloosterman sums, in some cases, these conjectures can be established unconditionally.

Metric decomposability theorems on sets of integers

AbstractA set is called additively decomposable (resp., asymptotically additively decomposable) if there exist sets of cardinality at least two each such that (resp., is finite). If none of these properties hold, the set is called totally primitive. We define ‐decomposability analogously with subsets of . Wirsing showed that almost all subsets of are totally primitive. In this paper, in the spirit of Wirsing, we study decomposability from a probabilistic viewpoint. First, we show that almost all symmetric subsets of are ‐decomposable. Then we show that almost all small perturbations of the set of primes yield a totally primitive set. Further, this last result still holds when the set of primes is replaced by the set of sums of two squares, which is by definition decomposable.

Observer-based control of polynomial fuzzy fractional-order systems

In this study, the Lyapunov technique is used to analyze the observer-based control problem for polynomial fuzzy fractional order (PFFO) models. The case of polynomial matrices with unmeasurable states is considered to increase the applicability of the PFFO models in the design problem. In this regard, we offer two design procedures. First, the design conditions are presented in a one-step procedure. In this design, the non-convex conditions are transformed in a set of sum of squares (SOS) by introducing a new symbolic variable except the state vector [Formula: see text] and its estimated [Formula: see text]. The obtained SOS conditions are presented involve three independent symbolic variables which increase both computing complexity and conservatism. To get around this shortcoming, a second design method is developed. By using the suggested method, SOS conditions requiring just two separate symbolic variables may be obtained. The architecture is shown in two stages; however, the observer and controller gains are computed in a single stage in order to further minimize conservatism. In order to demonstrate the utility of the suggested theoretical analysis, a simulated example is then provided.

Sign changes of fourier coefficients of holomorphic cusp forms at norm form arguments

AbstractLet f be a non-CM Hecke eigencusp form of level 1 and fixed weight, and let $\{\lambda_f(n)\}_n$ be its sequence of normalised Fourier coefficients. We show that if $K/ \mathbb{Q}$ is any number field, and $\mathcal{N}_K$ denotes the collection of integers representable as norms of integral ideals of K, then a positive proportion of the positive integers $n \in \mathcal{N}_K$ yield a sign change for the sequence $\{\lambda_f(n)\}_{n \in \mathcal{N}_K}$ . More precisely, for a positive proportion of $n \in \mathcal{N}_K \cap [1,X]$ we have $\lambda_f(n)\lambda_f(n') &lt; 0$ , where n′ is the first element of $\mathcal{N}_K$ greater than n for which $\lambda_f(n') \neq 0$ .For example, for $K = \mathbb{Q}(i)$ and $\mathcal{N}_K = \{m^2+n^2 \;:\; m,n \in \mathbb{Z}\}$ the set of sums of two squares, we obtain $\gg_f X/\sqrt{\log X}$ such sign changes, which is best possible (up to the implicit constant) and improves upon work of Banerjee and Pandey. Our proof relies on recent work of Matomäki and Radziwiłł on sparsely-supported multiplicative functions, together with some technical refinements of their results due to the author.In a related vein, we also consider the question of sign changes along shifted sums of two squares, for which multiplicative techniques do not directly apply. Using estimates for shifted convolution sums among other techniques, we establish that for any fixed $a \neq 0$ there are $\gg_{f,\varepsilon} X^{1/2-\varepsilon}$ sign changes for $\lambda_f$ along the sequence of integers of the form $a + m^2 + n^2 \leq X$ .

Metrological Aspects of Assessing Surface Topography and Machining Accuracy in Diagnostics of Grinding Processes

The paper presents probabilistic aspects of diagnostics of grinding processes with consideration of metrological aspects of evaluation of topography of machined surfaces and selected problems of assessment of machining accuracy. The processes of creating the geometric structure of the ground surface are described. It was pointed out that the distribution of features important for process diagnostics depends on the mechanism of cumulative effects of random disturbances. Usually, there is a multiplicative mechanism or an additive mechanism of the component vectors of relative displacements of the tool and workpiece. The paper describes a method for determining the classification ability of specific parameters used to evaluate stereometric features of ground surfaces. It is shown that the ability to differentiate the geometric structure of a certain set of surfaces using a selected parameter depends on the geometric mean of the differences in normalized and sorted, consecutive values of this parameter. A methodology is presented for evaluating the ability of various parameters to distinguish different geometric structures of surfaces. Further, on the basis of analyses of a number of grinding processes, a methodology was formulated for proceeding leading to a comprehensive evaluation of machining accuracy and forecasting its results. It was taken into account that in forecasting the accuracy of grinding, it is necessary to determine the deviations, arising under the conditions of multiplicative interaction of the effects of various causes of inaccuracy. Examples are given of processes in which, due to the deformation of the technological system, dependent on the position of the zone and machining force, varying temperature fields and tool wear, the distributions of dimensional deviations are not the realization of stationary processes. It was emphasized that on the basis of the characteristics of the dispersion of the deviation value in the sum set of elements, it is not possible to infer its causes. Only the determination of the "instantaneous" values of the deviation dispersion parameters allows a more complete diagnosis of the process.

On Min Sum Vertex Cover and Generalized Min Sum Set Cover

We study the Generalized Min Sum Set Cover (GMSSC) problem, wherein given a collection of hyperedges with arbitrary covering requirements , the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge is considered covered by the first time when and many of its vertices appear in the ordering. We give a approximation algorithm for GMSSC, coming close to the best possible bound of 4, already for the classical special case (with all ) of Min Sum Set Cover (MSSC) studied by Feige, Lovász, and Tetali, and improving upon the previous best known bound of due to Im, Sviridenko, and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Min Sum Vertex Cover (MSVC) is a well-known special case of MSSC in which the input hypergraph is a graph (i.e., ) and for every edge . We give a approximation for MSVC and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best approximation of Barenholz, Feige, and Peleg. Finally, we revisit MSSC and consider the norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of for all , giving another proof of the result of Golovin, Gupta, Kumar, and Tangwongsan, and showing its tightness up to NP-hardness. For , this gives yet another proof of the 4 approximation for MSSC.

Convexity, superquadratic growth, and dot products.

Let be a point set with cardinality . We give an improved bound for the number of dot products determined by , proving that A crucial ingredient in the proof of this bound is a new superquadratic expander involving products and shifts. We prove that, for any finite set , there exist such that This is derived from a more general result concerning growth of sets defined via convexity and sum sets, and which can be used to prove several other expanders with better than quadratic growth. The proof develops arguments from Hanson, Roche-Newton, and Rudnev [Combinatorica, to appear], and uses predominantly elementarymethods.

Some Spectral Characteristic Numbers of Direct Sum of Operators

In the present study, firstly, some algebraic inequalities are proved, which will be used later. By making use of these relations, some evaluations are found related the gaps between norm and numerical radius, spectral radius and Crawford number for diagonal block operator matrices on the infinite direct sum of Hilbert spaces. Later on, the gaps between some spectral characteristic numbers (operator norm, lower and upper bounds of spectrum set and numerical radius) of the infinite direct sum of Hilbert space operators relatively to the same spectral characteristics of the coordinate operators are investigated. Then, the obtained results are supported by applications. The open problem posed by Demuth in 2015 and the works of Kittaneh and his researcher group in this area had an important effect on the formation of the subject discussed in this study.