For a subset A of a field F, write A(A + 1) for the set {a(b + 1): a, b is an element of A). We establish new estimates on the size of A(A + 1) in the case where F is either a finite field of prime order, or the real ...
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For a subset A of a field F, write A(A + 1) for the set {a(b + 1): a, b is an element of A). We establish new estimates on the size of A(A + 1) in the case where F is either a finite field of prime order, or the real line. In the finite field case we show that A(A + 1) is of cardinality at least vertical bar A vertical bar(57/56-o(1)), so long as vertical bar A vertical bar < p(1/2). In the real case we show that the cardinality is at least vertical bar A vertical bar(24/19-o(1)). These improve on the previously best-known exponents of 106/105 - 0(1) and 5/4 respectively. (c) 2012 Elsevier Inc. All rights reserved.
A finite setA= {a(1)< horizontal ellipsis <a(n)}subset of Double-struck capital R is said to beconvexif the sequence (a(i)-a(i-1))i=2n is strictly increasing. Using an estimate of the additive energy of convex s...
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A finite setA= {a(1)< horizontal ellipsis additive energy of convex sets, one can estimate the size of the sumset as divide A+A divide greater than or similar to divide A divide (102/65), which slightly sharpens Shkredov's latest result divide A+A divide greater than or similar to divide A divide (58/37).
We show that pairs of generators for the family Sz(q) of Suzuki groups may be selected so that the corresponding Cayley graphs are expanders. By combining this with several deep works of Kassabov, Lubotzky and Nikolov...
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We show that pairs of generators for the family Sz(q) of Suzuki groups may be selected so that the corresponding Cayley graphs are expanders. By combining this with several deep works of Kassabov, Lubotzky and Nikolov, this establishes that the family of all non-abelian finite simple groups can be made into expanders in a uniform fashion.
We prove a new Elekes-Szabo type estimate on the size of the intersection of a Cartesian product Ax B x C with an algebraic surface if = 0} over the reals. In particular, if A, B, C are sets of N real numbers and f is...
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We prove a new Elekes-Szabo type estimate on the size of the intersection of a Cartesian product Ax B x C with an algebraic surface if = 0} over the reals. In particular, if A, B, C are sets of N real numbers and f is a trivariate polynomial, then either f has a special form that encodes additive group structure (for example, f (x, y, x) = x + y - z), or A x B x C pi if = 0} has cardinality O(N12/7). This is an improvement over the previous bound O(N11/6). We also prove an asymmetric version of our main result, which yields an Elekes-Ronyai type expanding polynomial estimate with exponent 3/2. This has applications to questions in combinatorial geometry related to the Erdos distinct distances problem. Like previous approaches to the problem, we rephrase the question as an L2 estimate, which can be analyzed by counting additive quadruples. The latter problem can be recast as an incidence problem involving points and curves in the plane. The new idea in our proof is that we use the order structure of the reals to restrict attention to a smaller collection of proximate additive quadruples.(c) 2023 Elsevier Inc. All rights reserved.
In this paper, we study combinatorial aspects of permutations of {1, ..., n} and related topics. In particular, we prove that there is a unique permutation pi of {1, ..., n} such that all the numbers k + (pi k) (k = 1...
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In this paper, we study combinatorial aspects of permutations of {1, ..., n} and related topics. In particular, we prove that there is a unique permutation pi of {1, ..., n} such that all the numbers k + (pi k) (k = 1, ..., n) are powers of two. We also show that n vertical bar per[i(j-1)](1 <= n, j <= n) for any integer n > 2. We conjecture that if a group G contains no element of order among 2, ..., n + 1 then any A subset of G with vertical bar A vertical bar = n can be written as {a1, ..., a(n)} with a(1), a(2)(2), ..., a(n)(n) pairwise distinct. This conjecture is confirmed when G is a torsion-free abelian group. We also prove that for any finite subset A of a torsion-free abelian group G with vertical bar A vertical bar = n > 3, there is a numbering a(1), ..., a(n) of all the elements of A such that all the n sums a(1) + a(2) + a(3), a(2 )+ a(3) + a(4), ..., a(n-2) + a(n-1) + a(n), a(n-1) + a(n) + a(1), a(n) + a(1) + a(2) are pairwise distinct, and conjecture that this remains valid if G is cyclic.
For (G, +) a finite abelian group the plus-minus weighted Davenport constant, denoted D +/-(G), is the smallest l such that each sequence g(1) ... g(l) over G has a weighted zerosubsum with weights + 1 and -1, i.e. th...
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For (G, +) a finite abelian group the plus-minus weighted Davenport constant, denoted D +/-(G), is the smallest l such that each sequence g(1) ... g(l) over G has a weighted zerosubsum with weights + 1 and -1, i.e. there is a non-empty subset I subset of {1, ..., l} such that Sigma(i is an element of I) a(i)g(i) = 0 for alpha(i) is an element of {+1, -1}. We present new bounds for this constant, mainly lower bounds, and also obtain the exact value of this constant for various additional types of groups.
We obtain a new sum-product estimate in prime fields for sets of large cardinality. In particular, we show that if Our argument builds on and improves some recent results of Shakan and Shkredov ['Breaking the 6/5 ...
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We obtain a new sum-product estimate in prime fields for sets of large cardinality. In particular, we show that if Our argument builds on and improves some recent results of Shakan and Shkredov ['Breaking the 6/5 threshold for sums and products modulo a prime', Preprint, 2018, arXiv:1806.07091v1] which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy of some subset. Our main novelty comes from reducing the estimation of to a point-plane incidence bound of Rudnev ['On the number of incidences between points and planes in three dimensions', Combinatorica 38(1) (2017), 219-254] rather than a point-line incidence bound used by Shakan and Shkredov.
In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and...
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In this paper, we prove some extensions of recent results given by Shkredov and Shparlinski on multiple character sums for some general families of polynomials over prime fields. The energies of polynomials in two and three variables are our main ingredients.
Suppose A subset of R of size k has distinct consecutive gamma-differences, that is for 1 >(r) vertical bar A vertical bar vertical bar B vertical bar 1/(r+1). Utilizing de Bruijn sequences, we show this inequality...
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Suppose A subset of R of size k has distinct consecutive gamma-differences, that is for 1 <= i <= k - r, the r-tuples (a(i+1) - ai,...,a(i+r) - a(i+r-1)) are distinct. Then for any finite B subset of R, one has vertical bar A + B vertical bar >>(r) vertical bar A vertical bar vertical bar B vertical bar 1/(r+1). Utilizing de Bruijn sequences, we show this inequality is sharp up to the constant. Moreover, for the sequence {n alpha}, a sharp upper bound for the size of the distinct consecutive r-differences is obtained, which generalizes Steinhaus' three gap theorem. A dual problem on the consecutive r-differences of the returning times for some phi is an element of R defined by {T : {T theta} < phi} is also considered, which generalizes a result of Slater. (C) 2018 Elsevier Inc. All rights reserved.
We show that if A subset of {1,..., N} has no solutions to a - b = n(2) with a, b is an element of A and n >= 1, then vertical bar A vertical bar 0. This improves upon a result of Pintz, Steiger, and Szemeredi.
We show that if A subset of {1,..., N} has no solutions to a - b = n(2) with a, b is an element of A and n >= 1, then vertical bar A vertical bar << N/(logN (c log log logN) for some absolute constant c > 0. This improves upon a result of Pintz, Steiger, and Szemeredi.
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