Sumset Research Articles

Sets of Sums of a Series Depending on Sign Distributions

Let (an) be a sequence of positive real numbers monotonically convergent to 0 for which ∑an diverges and let E be the set of sign distributions. We call $$S(E,{a_n}) = \{ \sum\limits_{n = 1}^\infty {{\varepsilon _n}{a_n}:({\varepsilon _n})} \in E\} $$ the set of E-sums for the sequence (an). In this paper we study topological properties of sets S(EUSA, an) and S(EBSA, an), where EUSA is the set of all uniform segmentally alternating sign distributions and EBSA is the family of all bounded segmentally alternating sign distributions.

Cauchy-Davenport Theorem for abelian groups and diagonal congruences

We prove an analogue of the Cauchy-Davenport Theorem and Chowla’s Theorem for sum sets in a general abelian group and give an application to diagonal congruences, establishing a best possible estimate for the distribution of solutions of a diagonal congruence ∑ i = 1 n a i x i k ≡ c ( mod q ) \sum _{i=1}^n a_ix_i^k \equiv c \pmod q with an arbitrary modulus.

Three-weight codes, triple sum sets, and strongly walk regular graphs

We construct strongly walk-regular graphs as coset graphs of the duals of three-weight codes over $$\mathbb {F}_q.$$ The columns of the check matrix of the code form a triple sum set, a natural generalization of partial difference sets. Many infinite families of such graphs are constructed from cyclic codes, Boolean functions, and trace codes over fields and rings. Classification in short code lengths is made for $$q=2,3,4$$ .

Superfractality of the Set of Incomplete Sums of One Positive Series

We consider a family of convergent positive normed series with real terms defined by the conditions where (an) is a nondecreasing sequence of real numbers. The structural properties of these series are investigated. For a special case, namely, $$ \left({a}_n\right)={2}^{n-1},{c}_n=\left(n+1\right){\tilde{r}}_n,n\in \mathbb{N} $$ , we study the geometry of the series (i.e., the properties of cylindrical sets, metric relations generated by them, and topological and metric properties of the set of all incomplete sums of the series). For the infinite Bernoulli convolution determined by this series, we describe its Lebesgue structure (discrete, absolutely continuous, and singular components) and spectral properties, as well as the behavior of the absolute value of the characteristic function at infinity. We also study finite self-convolutions of the distributions of this kind.

Exceptional set for sums of unlike powers of primes

Let [Formula: see text] denote the number of representations of an even integer [Formula: see text], [Formula: see text] as the sum of two squares, one cube and three fourth powers of primes, and by [Formula: see text] we denote the number of even integers [Formula: see text], [Formula: see text] such that the expected asymptotic formula for [Formula: see text] fails to hold. In this paper, it is proved that [Formula: see text] for any [Formula: see text].

Convex sequences may have thin additive bases

For a fixed $c > 0$ we construct an arbitrarily large set $B$ of size $n$ such that its sum set $B+B$ contains a convex sequence of size $cn^2$, answering a question of Hegarty.

Non-Weighted $L_2$ -Gain Control for Asynchronously Switched Linear Systems With Detectable Switching Instants and Ranged Mode-Identifying Time

In this paper, the non-weighted L 2 -gain control problem is addressed for a class of asynchronously switched linear systems, where the asynchronous phenomenon is caused by the mode-identifying process. Unlike the literature concerned with asynchronously switched systems, we construct a new class of clock-dependent Lyapunov function (CDLF), which can be permitted or prohibited to increase when the modes of the controller and system are unmatched. Furthermore, a novel controller design strategy is introduced. The asynchronous and synchronous controllers are designed separately, and are both clock-dependent. By using the CDLF approach, a clock-dependent sufficient condition characterizing the non-weighted L 2 -gain performance is obtained for the asynchronously switched systems. The controller gains can be computed by solving a set of sum of square (SOS) program. At last, the advantages of the results are illustrated within two examples.

On the few products, many sums problem

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, redistributing the exponents in the textbook Elekes sum-product inequality and the new best known additive energy bound $\mathsf E(A)\lesssim_M |A|^{49/20}$, which aligns, in a sense to be discussed, with the best known sum set bound $|A+A|\gtrsim_M |A|^{8/5}$. These bounds, with $M=1$, also apply to multiplicative subgroups of $\mathbb F^\times_p$, whose order is $O(\sqrt{p})$. We adapt the above energy bound to larger subgroups and obtain new bounds on gaps between elements in cosets of subgroups of order $\Omega(\sqrt{p})$.

Exceptional sets for sums of five and six almost equal prime cubes

We investigate the exceptional sets of natural number n which can be represented as sums of five and six cubes of almost equal primes, i.e. $${n=p_1^3+\cdots+p_s^3}$$ (s=5,6). It is established that almost all natural numbers n subject to certain congruence conditions have the above representation with $${|p_j-(n/s)^{\frac{1}{3}}|\leq n^{\theta_s/3+\varepsilon}}$$ ( $${1\leq j\leq s}$$ ), where $${\theta_5=8/9+\varepsilon}$$ and $${\theta_6=5/6+\varepsilon}$$ .

Formula omitted]-recurrence and nilsystems

By a result due to Furstenberg, a homeomorphism T of a compact space is distal if and only if it possesses the property of IP⁎-recurrence, meaning that for any x0∈X, for any open neighborhood U of x0, and for any sequence (ni) in Z, the set RU(x0)={n∈Z:Tnx0∈U} has a non-trivial intersection with the set of finite sums {ni1+ni2+⋯+nis:i1<i2<…<is,s∈N}. We show that translations on compact nilmanifolds (which are known to be distal) are characterized by a stronger property of IPr⁎-recurrence, which asserts that for any x0∈X and any neighborhood U of x0 there exists r∈N such that for any r-element sequence n1,…,nr in Z the set RU(x0) has a non-trivial intersection with the set {ni1+ni2+⋯+nis:i1<i2<…<is,s≤r}. We also show that the property of IPr⁎-recurrence is equivalent to an ostensibly much stronger property of polynomial IPr⁎-recurrence. (This should be juxtaposed with the fact that for general distal transformations the polynomial IP⁎-recurrence is strictly stronger than the IP⁎-recurrence.)

Towards the classification of finite simple groups with exactly three or four supercharacter theories

A supercharacter theory for a finite group [Formula: see text] is a set of superclasses each of which is a union of conjugacy classes together with a set of sums of irreducible characters called supercharacters that together satisfy certain compatibility conditions. The aim of this paper is to give a description of some finite simple groups with exactly three or four supercharacter theories.

On minimal asymptotic basis of order [formula omitted

Let g⩾3 be an integer. Let A be the set of all nonempty finite sums of distinct powers of g and all numbers of the form 2⋅gr with r⩾0. In this paper, we construct an asymptotic basis A of order g−1 no subset of which is minimal. The case g=3 was obtained by Nathanson.

The geometry of Gaussian integer continued fractions

The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship |ad−bc|=1 for a,b,c,d integers as it arises in the Farey tessellation of the hyperbolic plane H2 and the array of Ford circles in the upper-half of the real plane R2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition |αγ−βδ|=1 for α,β,γ,δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers.

A Novel Nested Configuration Based on the Difference and Sum Co-Array Concept.

Recently, the concept of the difference and sum co-array (DSCa) has attracted much attention in array signal processing due to its high degree of freedom (DOF). In this paper, the DSCa of the nested array (NA) is analyzed and then an improved nested configuration known as the diff-sum nested array (DsNA) is proposed. We find and prove that the sum set for the NA contains all the elements in the difference set. Thus, there exists the dual characteristic between the two sets, i.e., for the difference result between any two sensor locations of the NA, one equivalent non-negative/non-positive sum result of two other sensor locations can always be found. In order to reduce the redundancy for further DOF enhancement, we develop a new DsNA configuration by moving nearly half the dense sensors of the NA to the right side of the sparse uniform linear array (ULA) part. These moved sensors together with the original sparse ULA form an extended sparse ULA. For analysis, we provide the closed form expressions of the DsNA locations as well as the DOF. Compared with some novel sparse arrays with large aperture such as the NA, coprime array and augmented nested array, the DsNA can achieve a higher number of DOF. The effectiveness of the proposed array is proved by the simulations.

Divisibility problems for function fields

We investigate three combinatorial problems considered by Erdős, Rivat, Sarkozy and Schon regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.

On an alternative, implicit renormalization procedure for the Casimir energy

We propose a procedure for the renormalization of Casimir energy that is based on the implicit versions of standard steps of renormalization procedure — regularization, subtraction and removing the regularization. The proposed procedure is based on the calculation of a set of convergent sums, which are related to the original divergent sum for the non-renormalized Casimir energy. Then, we construct a linear equation system that connects this set of convergent sums with the renormalized Casimir energy, which is a solution to this system of equations. This procedure slightly reduces the indeterminacy that arises in the standard procedure when we choose the particular regularization and the explicit form of the counterterm. The proposed procedure can be applied not only to systems with the explicit transcendental equation for the spectrum but also to systems with the spectrum that can be obtained only numerically. However, to perform the proposed procedure, we need a parameter of the system that satisfies two conditions: (i) we can obtain explicit analytical expressions (as a function of our parameter) for coefficients for all divergent and unphysical terms in divergent sum for Casimir energy; (ii) infinite value of this parameter should be the “natural” renormalization point, i.e. Casimir energy must tend to zero when parameter tends to infinity.

On the Possibility of the Implicit Renormalization of the Casimir Energy

We propose a procedure for renormalizing the Casimir energy that makes the steps that are used in the standard renormalization procedure, that is, regularization, subtraction, and deregularization, implicit. The proposed procedure is based on the calculation of a set of convergent sums, each of which is related to the initial divergent sum of the non-renormalized Casimir energy. Next, we construct a system of linear equations that relates this set of convergent sums to the renormalized Casimir energy. The unknown renormalized Casimir energy is obtained as a result of solving this system of equations. In this case, both the calculations of the convergent sums and the subsequent solution of the system of linear equations are performed with a certain (generally speaking, arbitrary) ordered accuracy; thus, the result is also approximate. The proposed procedure is, first, more computationally effective than the standard one, and, second, applicable not only to the problems where a transcendental equation for the spectrum can be written, but also to the problems where the spectrum is known only numerically.

On the cardinality of subsequence sums

Let [Formula: see text] be a finite sequence of integers, where [Formula: see text] and [Formula: see text] with [Formula: see text] for [Formula: see text]. A subsequence sum of [Formula: see text] is the sum of all terms of a nonempty subsequence of [Formula: see text]. Denoted by [Formula: see text] the set of all subsequence sums of [Formula: see text]. In this paper, for given [Formula: see text], we give the lower bound for [Formula: see text] with in terms of [Formula: see text] and the numbers of positive, negative integers in [Formula: see text]. We also determine the structure of the finite sequence [Formula: see text] of integers for which [Formula: see text] is minimal. This generalizes the results of Mistri, Pandey and Prakash. Moreover, we give a correction to a result of their paper.

Full Dimensional Sets of Reals Whose Sums of Partial Quotients Increase in Certain Speed

For a real x∈(0,1)∖Q, let x=[a1(x),a2(x),⋯] be its continued fraction expansion. Let sn(x)=∑j=1naj(x). The Hausdorff dimensions of the level setsEφ(n),α:={x∈(0,1):limn→∞⁡sn(x)φ(n)=α} for α≥0 and a non-decreasing sequence {φ(n)}n=1∞ have been studied by E. Cesaratto, B. Vallée, J. Wu, J. Xu, G. Iommi, T. Jordan, L. Liao, M. Rams et al. In this work we carry out a kind of inverse project of their work, that is, we consider the conditions on φ(n) under which one can expect a 1-dimensional set Eφ(n),α. We give certain upper and lower bounds on the increasing speed of φ(n) when Eφ(n),α is of Hausdorff dimension 1 and a new class of sequences {φ(n)}n=1∞ such that Eφ(n),α is of full dimension. For an irregular sequence {φ(n)}n=1∞, a full dimensional set Eφ(n),α is impossible.

A Novel Noncircular MUSIC Algorithm Based on the Concept of the Difference and Sum Coarray.

In this paper, we propose a vectorized noncircular MUSIC (VNCM) algorithm based on the concept of the coarray, which can construct the difference and sum (diff–sum) coarray, for direction finding of the noncircular (NC) quasi-stationary sources. Utilizing both the NC property and the concept of the Khatri–Rao product, the proposed method can be applied to not only the ULA but also sparse arrays. In addition, we utilize the quasi-stationary characteristic instead of the spatial smoothing method to solve the coherent issue generated by the Khatri–Rao product operation so that the available degree of freedom (DOF) of the constructed virtual array will not be reduced by half. Compared with the traditional NC virtual array obtained in the NC MUSIC method, the diff–sum coarray achieves a higher number of DOFs as it comprises both the difference set and the sum set. Due to the complementarity between the difference set and the sum set for the coprime array, we choose the coprime array with multiperiod subarrays (CAMpS) as the array model and summarize the properties of the corresponding diff–sum coarray. Furthermore, we develop a diff–sum coprime array with multiperiod subarrays (DsCAMpS) whose diff–sum coarray has a higher DOF. Simulation results validate the effectiveness of the proposed method and the high DOF of the diff–sum coarray.