Long arithmetic progressions in sumsets: Thresholds and bounds

Szemerédi, Endre; Vu, Van

doi:10.1090/s0894-0347-05-00502-3

Cited by 35 publications

(39 citation statements)

References 27 publications

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“…(The length of an arithmetic progression is the number of its terms, less 1). As indicated in [SV06], "many estimates for f (n, k, l) have been discovered by Bourgain, Freiman, Halberstam, Green, Ruzsa, and Sárközy". It is worth noting in this connection that Theorem 2 establishes the exact value of this function for k large; namely, it is easy to deduce from Theorem 2 (and keeping in mind the trivial example…”

Section: Background and Summary Of Resultsmentioning

confidence: 98%

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Consecutive integers in high-multiplicity sumsets

Lev

2010

Acta Math Hung

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Abstract. Sharpening (a particular case of) a result of Szemerédi and Vu [SV06] and extending earlier results of Sárközy [S89] and ourselves [L97b], we find, subject to some technical restrictions, a sharp threshold for the number of integer sets needed for their sumset to contain a block of consecutive integers of length, comparable with the lengths of the set summands.A corollary of our main result is as follows. Let k, l 1 and n 3 be integers, and suppose that A 1 , . . . , A k ⊆ [0, l] are integer sets of size at least n, none of which is contained in an arithmetic progression with difference greater than 1. If k 2 ⌈(l − 1)/(n − 2)⌉, then the sumset A 1 +· · ·+A k contains a block of consecutive integers of length k(n − 1).

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Section: Background and Summary Of Resultsmentioning

confidence: 98%

“…Though the proof of Theorem 3, presented in [SV06], is constructive, the constants C and c are not computed explicitly. Indeed, the argument leads to excessively large values of these constants, which may present a problem in some applications.…”

Section: Background and Summary Of Resultsmentioning

confidence: 99%

Consecutive integers in high-multiplicity sumsets

Lev

2010

Acta Math Hung

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“…Theorem 3.3 is an existential result about an arithmetic progression (of stepsize 1) in the set of all subset sums, and thus belongs to the realm of additive combinatorics, see [47] for an overview. Arithmetic progressions in the set of all subsets sums have been studied at least since early work of Alon [4], see also the literature by Erdős, Freiman, Sárközy, Szemerédi, and others [5,21,23,33,34,35,42,43,46]. A result of this type is also implicit in the algorithm by Galil and Margalit [25], but only applies to sets.…”

Section: Structural Partmentioning

confidence: 99%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.

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“…This is to be expected since there are lots of collisions when forming subset sums of {a 1 , • • • , a d } when d ≥ p 1/8+ε , and thus H should have rich additive structures. This type of phenomenon from subset sums or iterated sumsets was studied in [13,14]. Then we use character sum estimates to show that P must contain primitive roots; see Proposition 3.2 below.…”

Section: Previously the Best Known Upper Bound Is F (P)mentioning

confidence: 99%

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

Alsetri¹,

Shao²

2021

Preprint

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We consider the problem of bounding the dimension of Hilbert cubes in a finite field Fp that does not contain any primitive roots. We show that the dimension of such Hilbert cubes is Oε(p 1/8+ε ) for any ε > 0, matching what can be deduced from the classical Burgess estimate in the special case when the Hilbert cube is an arithmetic progression. We also consider the dual problem of bounding the dimension of multiplicative Hilbert cubes avoiding an interval.

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Long arithmetic progressions in sumsets: Thresholds and bounds

Abstract: For a set A A of integers, the sumset l A = A + ⋯ + A lA =A+\dots +A consists of those numbers which can be represented as a sum of l l elements of A A : \[ l A = { a 1 + ⋯ + a l | a i ∈ A i … Show more

Cited by 35 publications

References 27 publications

Consecutive integers in high-multiplicity sumsets

Consecutive integers in high-multiplicity sumsets

On Near-Linear-Time Algorithms for Dense Subset Sum

Upperbound for Dimension of Hilbert Cubes contained in the Quadratic Residues of $\mathbb{F_p}$

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