Roth’s theorem in ℤ4n

Sanders, Tom

doi:10.2140/apde.2009.2.211

Cited by 12 publications

(15 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Meshulam's [40] long-standing bound r 3 (Z n m ) = O( m n n ) for odd values of m ≥ 3 was extended by Lev [38] to even values m ≥ 4. Improving this, Sanders [52] proved the following result:…”

Section: Introductionmentioning

confidence: 88%

“…Meshulam’s [ 40 ] long-standing bound for odd values of was extended by Lev [ 38 ] to even values . Improving this, Sanders [ 52 ] proved the following result: for some positive c . Green and Tao [ 33 ] write that is admissible.…”

Section: Introductionmentioning

confidence: 99%

“…For a lower bound when m = 4 Sanders [52] proved: there exists S ⊂ G = (Z n 4 , +) which does not contain a proper three term arithmetic progression with [24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Elsholtz

Pach

2020

Des. Codes Cryptogr.

View full text Add to dashboard Cite

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).

show abstract

Section: Introductionmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Elsholtz

Pach

2020

Des. Codes Cryptogr.

View full text Add to dashboard Cite

show abstract

“…Why were they looking at the Z/4Z question? Lev was kind enough to provide an answer (personal communication, 2018): their initial motivation was to improve Sanders's result "Roth's Theorem in (Z/4Z) n " [111]. Very shortly thereafter, and nearly simultaneously with one another, Ellenberg and Gijswijt leveraged the Croot-Lev-Pach technique to give a positive resolution to the Cap Set Conjecture [39].…”

Section: Definition 13 (Freiman Homomorphism) a Freiman K-homomorphmentioning

confidence: 99%

New applications of the polynomial method: The cap set conjecture and beyond

Grochow

2018

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

The cap set problem asks how large can a subset of (Z/3Z) n be and contain no lines or, more generally, how can large a subset of (Z/pZ) n be and contain no arithmetic progressions. This problem was motivated by deep questions about structure in the prime numbers, the geometry of lattice points, and the design of statistical experiments. In 2016, Croot, Lev, and Pach solved the analogous problem in (Z/4Z) n , showing that the largest set without arithmetic progressions had size at most c n for some c < 4. Their proof was as elegant as it was unexpected, being a departure from the tried and true methods of Fourier analysis that had dominated the field for half a century. Shortly thereafter, Ellenberg and Gijswijt leveraged their method to resolve the original cap set problem. This expository article covers the history and motivation for the cap set problem and some of the many applications of the technique: from removing triangles from graphs, to rigidity of matrices, and to algorithms for matrix multiplication. The latter application turns out to give back to the original problem, sharpening our understanding of the techniques involved and of what is needed for showing that the current bounds are tight. Most of our exposition assumes only familiarity with basic linear algebra, polynomials, and the integers modulo N .

show abstract

“…Ясно, что понятие арифметической прогрессии длины три или, иными сло-вами, трех точек x, y, z, удовлетворяющих уравнению x + y = 2z, бессмысленно в группе Z n 2 . Поэтому простейшей из групп Z n p , в которой можно рассмотреть задачу Рота, является группа Z n 3 (другие абелевы группы изучались в статьях [93]- [95]). …”

unclassified

Анализ Фурье В Комбинаторной Теории Чисел

Шкредов¹,

Shkredov²

2010

Uspekhi Mat. Nauk

View full text Add to dashboard Cite

Roth’s theorem in ℤ₄ⁿ

Cited by 12 publications

References 21 publications

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

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

New applications of the polynomial method: The cap set conjecture and beyond

Анализ Фурье В Комбинаторной Теории Чисел

Contact Info

Product

Resources

About