We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, ...
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We obtain a polynomial criterion for a set to have a small doubling in terms of the common energy of its subsets. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size ...
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Two sets of 0-1 vectors of fixed length form a uniquely decodeable code pair if their Cartesian product is of the same size as their sumset, where the addition is pointwise over integers. For the size of the sumset of...
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Two sets of 0-1 vectors of fixed length form a uniquely decodeable code pair if their Cartesian product is of the same size as their sumset, where the addition is pointwise over integers. For the size of the sumset of such a pair, van Tilborg has given an upper bound in the general case. Urbanke and Li, and later Ordentlich and Shayevitz, have given better bounds in the unbalanced case, that is, when either of the two sets is sufficiently large. Improvements to the latter bounds are presented.
In the present paper we show that if A is a set of n real numbers and the product set A. A has at most n(1+epsilon) elements, then the h-fold sumset hA has at least n(log(h/2)/2) (log 2+1/2-fh(epsilon)) elements, wher...
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In the present paper we show that if A is a set of n real numbers and the product set A. A has at most n(1+epsilon) elements, then the h-fold sumset hA has at least n(log(h/2)/2) (log 2+1/2-fh(epsilon)) elements, where f(h)(c) -> 0 as c -> 0. We also prove results on the h-fold sumset h(A.A) = A.A + ... + A.A.
Let A, B be invertible n x n matrices over the finite field F-q with irreducible characteristic polynomials. For k is an element of Z(+) denote M-k(A, B) := {f (A)g(B): f, g is an element of F-q[x] with deg f, deg g =...
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Let A, B be invertible n x n matrices over the finite field F-q with irreducible characteristic polynomials. For k is an element of Z(+) denote M-k(A, B) := {f (A)g(B): f, g is an element of F-q[x] with deg f, deg g < k} Assume q >= 2n, we prove that vertical bar M-k(A, B)vertical bar > 4(q)(-k)q(min(n,2k-1)). Moreover, let d = dim Ker(AB - BA), we prove vertical bar M-k(A, B)vertical bar > 1/16((n)(d))(2(n-d))(n-d/2) q(k+min(k/2, n-d/2)) (c) 2013 Elsevier Inc. All rights reserved.
We consider the so called Magnus-Derek game, which is a two-person game played on a round table with n positions. The two players are called Magnus and Derek. Initially there is a token placed at position O. In each r...
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We consider the so called Magnus-Derek game, which is a two-person game played on a round table with n positions. The two players are called Magnus and Derek. Initially there is a token placed at position O. In each round Magnus chooses a positive integer m <= n/2 as the distance of the targeted position from his current position for the token to move, and Derek decides a direction, clockwise or counterclockwise, to move the token. The goal of Magnus is to maximize the total number of positions visited, while Derek's is to minimize this number. If both players play optimally, we prove that Magnus, the maximizer, can achieve his goal in O(n) rounds, which improves a previous result with O(n log n) rounds. Then we consider a modified version of the Magnus-Derek game, where one of the players reveals his moves in advance and the other player plays optimally. In this case we prove that it is NP-hard for Derek to achieve his goal if Magnus reveals his moves in advance. On the other hand, Magnus has an advantage to occupy all positions. We also consider the circumstance that both players play randomly, and we show that the expected time to visit all positions is O(n log n). (c) 2010 Elsevier B.V. All rights reserved.
As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic va...
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As a crucial technique for integrated circuits (IC) test response compaction, X-compact employs a special kind of codes called X-codes for reliable compressions of the test response in the presence of unknown logic values (Xs). From a combinatorial view point, Fujiwara and Colbourn introduced an equivalent definition of X-codes and studied X-codes of small weights that have good detectability and X-tolerance. An (m, n, d, x) X-code is an mxn binary matrix with column vectors as its codewords. The parameters d, x correspond to the test quality of the code. In this paper, bounds and constructions for constant weighted X-codes are investigated. First, we obtain a general result on the maximum number of codewords n for an (m, n, d, x) X-code of weight w, and we further improve this lower bound for the case with x = 2 and w = 3 through the probabilistic method. Then, using tools from additive combinatorics and finite fields, we present some explicit constructions for constant weighted Xcodes with d = 3, 7 and x = 2, which are optimal for the case when d = 3, w = 4 and nearly optimal for the case when d = 3, w = 3. We also consider a special class of X-codes introduced by Fujiwara and Colbourn and improve the best known lower bound on the maximum number of codewords for this kind of X-codes.
Approximate algebraic structures play a defining role in additive number theory and have found remarkable applications to questions in theoretical computer science, including in pseudorandomness and probabilistically ...
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Approximate algebraic structures play a defining role in additive number theory and have found remarkable applications to questions in theoretical computer science, including in pseudorandomness and probabilistically checkable proofs. Here we study approximate representations of finite groups: functions psi : G -> U-d such that Pr[psi(xy) = psi(x) psi(y)] is large or, more generally, such that the expected l(2) norm squared E-x,E-y parallel to psi(xy) - psi(x) psi(y)parallel to(2)(2) is small, where x, y are uniformly random elements of the group G and U-d denotes the group of unitary operators on C-d. We bound these quantities in terms of the ratio d/d(min) where d(min) is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. These results demonstrate that if G is quasi-random in the sense of Gowers, that is, if dmin is large, then G cannot be embedded in a small number of dimensions, or in a less-quasi-random group, without significant distortion of G's multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations.
The degrees-of-freedom of a user Gaussian interference channel (GIC) has been defined to be the multiple of (1/2)log(2) P at which the maximum sum of achievable rates grows with increasing power P. In this paper, we e...
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The degrees-of-freedom of a user Gaussian interference channel (GIC) has been defined to be the multiple of (1/2)log(2) P at which the maximum sum of achievable rates grows with increasing power P. In this paper, we establish that the degrees-of-freedom of three or more user, real, scalar GICs, viewed as a function of the channel coefficients, is discontinuous at points where all of the coefficients are nonzero rational numbers. More specifically, for all K > 2, we find a class of user GICs that is dense in the GIC parameter space for which K/2 degrees-of-freedom are exactly achievable, and we show that the degrees-of-freedom for any GIC with nonzero rational coefficients is strictly smaller than K/2. These results are proved using new connections with number theory and additive combinatorics.
We have formalised Szemeredi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For...
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We have formalised Szemeredi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.
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