Improved bounds for progression-free sets in $$C_8^{n}$$

Petrov, Fedor; Pohoata, Cosmin

doi:10.1007/s11856-020-1977-0

Cited by 9 publications

(11 citation statements)

References 13 publications

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“…(A version of this result, in the special case has also been observed in [ 44 ], having seen a precursor of this manuscript. Their main concern is an improvement of the upper bound.)…”

Section: Results and Methodssupporting

confidence: 67%

“…As J (s) is decreasing and J (3) ≤ 0.9184, with the additional consideration of composite m (see below), one can conclude, that for every m ≥ 3 the following holds (see e.g. [6,44]).…”

Section: Remark 11mentioning

confidence: 94%

“…As J ( s ) is decreasing and , with the additional consideration of composite m (see below), one can conclude, that for every the following holds (see e.g. [ 6 , 44 ]). For m not being a power of 2 this also holds: if p is an odd prime divisor of m , then .…”

Section: Introductionmentioning

confidence: 99%

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Caps and progression-free sets in $${{\mathbb {Z}}}_m^n$$

Elsholtz

Pach

2020

Des. Codes Cryptogr.

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We study progression-free sets in the abelian groups $$G=({{\mathbb {Z}}}_m^n,+)$$ G = ( Z m n , + ) . Let $$r_k({{\mathbb {Z}}}_m^n)$$ r k ( Z m n ) denote the maximal size of a set $$S \subset {{\mathbb {Z}}}_m^n$$ S ⊂ Z m n that does not contain a proper arithmetic progression of length k. We give lower bound constructions, which e.g. include that $$r_3({{\mathbb {Z}}}_m^n) \ge C_m \frac{((m+2)/2)^n}{\sqrt{n}}$$ r 3 ( Z m n ) ≥ C m ( ( m + 2 ) / 2 ) n n , when m is even. When $$m=4$$ m = 4 this is of order at least $$3^n/\sqrt{n}\gg \vert G \vert ^{0.7924}$$ 3 n / n ≫ | G | 0.7924 . Moreover, if the progression-free set $$S\subset {{\mathbb {Z}}}_4^n$$ S ⊂ Z 4 n satisfies a technical condition, which dominates the problem at least in low dimension, then $$|S|\le 3^n$$ | S | ≤ 3 n holds. We present a number of new methods which cover lower bounds for several infinite families of parameters m, k, n, which includes for example: $$r_6({{\mathbb {Z}}}_{125}^n) \ge (85-o(1))^n$$ r 6 ( Z 125 n ) ≥ ( 85 - o ( 1 ) ) n . For $$r_3({{\mathbb {Z}}}_4^n)$$ r 3 ( Z 4 n ) we determine the exact values, when $$n \le 5$$ n ≤ 5 , e.g. $$r_3({{\mathbb {Z}}}_4^5)=124$$ r 3 ( Z 4 5 ) = 124 , and for $$r_4({{\mathbb {Z}}}_4^n)$$ r 4 ( Z 4 n ) we determine the exact values, when $$n \le 4$$ n ≤ 4 , e.g. $$r_4({{\mathbb {Z}}}_4^4)=128$$ r 4 ( Z 4 4 ) = 128 . With regard to affine caps, i.e. sets without 3 points on a line, the new methods asymptotically improve the known lower bounds, when $$m=4$$ m = 4 and $$m=5$$ m = 5 : in $${{\mathbb {Z}}}_4^n$$ Z 4 n from $$2.519^n$$ 2 . 519 n to $$(3-o(1))^n$$ ( 3 - o ( 1 ) ) n , and when $$m=5$$ m = 5 from $$2.942^n$$ 2 . 942 n to $$(3-o(1))^n$$ ( 3 - o ( 1 ) ) n . This last improvement modulo 5 appears to be the first asymptotic improvement of any cap in AG(n, m), when $$m \ge 5$$ m ≥ 5 over a tensor lifting from dimension 6 (see Edel, in Des Codes Crytogr 31:5–14, 2004).

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“…(A version of this result, in the special case has also been observed in [ 44 ], having seen a precursor of this manuscript. Their main concern is an improvement of the upper bound.)…”

Section: Results and Methodssupporting

confidence: 67%

“…As J (s) is decreasing and J (3) ≤ 0.9184, with the additional consideration of composite m (see below), one can conclude, that for every m ≥ 3 the following holds (see e.g. [6,44]).…”

Section: Remark 11mentioning

confidence: 94%

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Caps and progression-free sets in $${{\mathbb {Z}}}_m^n$$

Elsholtz

Pach

2020

Des. Codes Cryptogr.

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“…A few years later Szemerédi [19] settled the former conjecture in the affirmative. The result of Erdős and Heilbronn for F p and Szemerédi's theorem for general groups were both later refined by Olson in [14], [15] and [16], who proved that OL(G) ≤ 2 |G| and also introduced a remarkable group ring approach (which has also recently resurfaced in the context of the polynomial method developments around the cap set problem; see [17] and [18]). Olson's result was subsequently pushed further by Hamidoune and Zemor [9], who proved that OL(G) ≤ 2|G| + O(|G| 1/3 log |G|), and among other things established the correct order of growth for OL(F p ), up to lower order terms.…”

Section: Introductionmentioning

confidence: 99%

Zero subsums in vector spaces over finite fields

Pohoata,

Zakharov

2020

Preprint

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The Olson constant OL(F d p ) represents the minimum positive integer t with the property that every subset A ⊂ F d p of cardinality t contains a nonempty subset with vanishing sum. The problem of estimating OL(F d p ) is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case d = 1.In this paper, we prove that for any fixed d ≥ 2 and ǫ > 0, the Olson constant of F d p satisfies the inequality OL(F d p ) ≤ (d − 1 + ǫ)p for all sufficiently large primes p. This settles a conjecture of Hoi Nguyen and Van Vu.

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“…Their proof uses a new polynomial method developed by Croot, Lev, and Pach for the analogous problem in

Z_{4}^{n}

. The preprint of Ellenberg and Gijswijt appeared just a few weeks after the one of Croot, Lev, and Pach, and subsequently a lot more activity evolved around these new ideas (see ).…”

Section: Introductionmentioning

confidence: 99%

A lower bound for the k‐multicolored sum‐free problem in Zmn

Lovász

Sauermann

2018

Proc. London Math. Soc.

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In this paper, we give a lower bound for the maximum size of a k‐colored sum‐free set in Zmn, where k⩾3 and m⩾2 are fixed and n tends to infinity. If m is a prime power, this lower bound matches (up to lower order terms) the previously known upper bound for the maximum size of a k‐colored sum‐free set in Zmn. This generalizes a result of Kleinberg–Sawin–Speyer for the case k=3 and as part of our proof we also generalize a result by Pebody that was used in the work of Kleinberg–Sawin–Speyer. Both of these generalizations require several key new ideas.

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Improved bounds for progression-free sets in $$C_8^{n}$$

Cited by 9 publications

References 13 publications

Caps and progression-free sets in $${{\mathbb {Z}}}_m^n$$

Caps and progression-free sets in $${{\mathbb {Z}}}_m^n$$

Zero subsums in vector spaces over finite fields

A lower bound for the k‐multicolored sum‐free problem in Zmn

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