On almost Cap sets in three variables and the multivariable Cap set problem
OON ALMOST CAP SETS IN THREE VARIABLES AND THE MULTIVARIABLE CAPSET PROBLEM
ALEXANDER FISH AND DIBYENDU ROY
Abstract.
In this note we prove that almost cap sets A ⊂ F nq , i.e., the subsets of F nq that do not containtoo many arithmetic progressions of length three, satisfy that | A | < c nq for some c q < q . As a corollary weprove a multivariable analogue of Ellenberg-Gijswijt theorem [3]. Introduction
We denote by F q the field with q elements. Let a, b, c ∈ F q \ { } be such that a + b + c = 0. A set A ⊂ F nq is called a cap set (for the tuple ( a, b, c )) if any solution ( x, y, z ) ∈ A of the equation ax + by + cz = 0 isof the form x = y = z . In a breakthrough paper Lev-Croot-Pach [2] showed that for n large enough, anycap set A ⊂ Z n for the tuple (1 , , −
2) satisfies that | A | ≤ r n , for some r <
4. Extending their method,Ellenberg-Gijswijt [3] showed that for any q there exists c q < q such that any cap set A ⊂ F nq satisfies | A | ≤ c nq .In this paper we obtain analogous upper bounds on the cardinality of almost cap sets in F nq . These setssatisfy a much weaker structural rigidity than cap sets. We say that a set A ⊂ F nq is an ( (cid:15), δ ) − cap set for (cid:15), δ > a, b, c ) ∈ ( F q \ { } ) with a + b + c = 0, if there exists A (cid:48) ⊂ A with | A (cid:48) | > δ | A | andsuch that for every x ∈ A (cid:48) the number of pairs ( y, z ) ∈ A satisfying ax + by + cz = 0 is less than | A | (cid:15) .Our main result gives an upper bound on the maximal cardinality of an ( (cid:15), δ ) − cap set A ⊂ F nq . For anychoice of (cid:15) > a = ( a, b, c ) ∈ ( F q \ { } ) satisfying a + b + c = 0, and a set A ⊂ F nq we denote by A (cid:15) a thefollowing set: A (cid:15) a = { x ∈ A | ∃ at least | A | (cid:15) pairs ( y, z ) ∈ A with ax + by + cz = 0 } . Theorem 1.1.
There exist (cid:15) > and c q < q such that for any δ > , A ⊂ F nq with | A | > c nq and a = ( a, b, c ) ∈ ( F q \ { } ) for sufficiently large n we have | A (cid:15) a | ≥ (1 − δ ) | A | . In other words, there exists (cid:15) > such that for any δ > , the ( (cid:15), δ ) − cap sets A ⊂ F nq satisfy that | A | ≤ c nq for sufficiently large n . To prove Theorem 1.1 we use Tao’s symmetric reformulation [5] of the method of Croot-Lev-Pach-Ellenberg-Gijswijt together with Lovett’s lower bound on the slice rank of tensors in terms of the cardinality of anindependent set [4].We also extend the Ellenberg-Gijswijt [3] upper bound on the cardinality of cap sets to the case of morethan three variables.
Date : February 24, 2021.
Key words and phrases.
Cap sets problem, arithmetic progressions. The notion of the cap set naturally extends to modules over rings. a r X i v : . [ m a t h . N T ] F e b ALEXANDER FISH AND DIBYENDU ROY
Theorem 1.2.
Let a , . . . , a d ∈ F q \ { } for d ≥ be such that a + . . . + a d = 0 . There exists c q < q suchthat for n sufficiently large, any set A ⊂ F nq with | A | > c nq contains all distinct x , . . . , x d satisfying (1) a x + a x + . . . + a d x d = 0 . Acknowledgements.
The authors thank the Australian Mathematical Sciences Institute (AMSI) for theirsupport of the second author through the Vacation Research Scholarship Program. The first author wassupported by the Australian Research Council grant DP210100162.2.
Slice rank of d -tensors versus Tao slice rank of functions on A d Let A ⊂ F nq , and F : A d → F q be a function for d ≥
2. Following Tao in [5], we define the Tao slice rank of F , denoted as T-srank( F ), as follows:If there exist functions f : A → F q , g : A d − → F q and i ∈ { , . . . , d } such that F ( x , . . . , x d ) = f ( x i ) g ( x i ),where x i is the ( d − x , . . . , x d ) by removing x i , then we define that T-srank( F ) = 1.We define that T-srank( F ) ≤ k for k ≥ F , . . . , F k : A d → F q of Tao slice rankequal to 1, and such that F = (cid:80) kj =1 F j .Denote by V a finite dimensional vector space over F q . Any multilinear function T : V d → F q is called a d -tensor. The slice rank of a d -tensor T , denoted as s-rank( T ), is defined as follows:We define that s-rank(T) = 1 if there exist 1-tensor T : V → F q and ( d − T : V d − → F q ,and i ∈ { , . . . , d } such that T ( v , . . . , v d ) = T ( v i ) T ( v i ), where v i ∈ V d − obtained by removing v i from( v , . . . , v d ), for any ( v , . . . , v d ) ∈ V d . We define that s-rank( T ) ≤ k for k ≥ k d -tensors T , . . . , T k : V d → F q of the slice rank equal to 1 such that T = (cid:80) ki =1 T i .For every F : A d → F q we correspond the d -tensor T F on the space V of all functions from A to F q definedas follows:(2) T F ( f , . . . , f d ) := (cid:88) ( x ,...,x d ) ∈ A d F ( x , . . . , x d ) f ( x ) . . . f d ( x d ) . The next lemma follows immediately from the definitions of the slice rank of d -tensors, the Tao slice rank offunctions on A d and the relationship between F and T F . Lemma 2.1. s-rank( T F ) ≤ T-srank( F ) . Assume that T is a d -tensor on the space V = F Nq . By multilinearity of T we have: T ( x , . . . , x d ) = (cid:88) α ∈ [ N ] d c α x α . . . x dα d , for ( x , . . . , x d ) ∈ V d , where any vector v ∈ V is represented in the coordinates as v = ( v , . . . , v N ), and α ∈ [ N ] d = { , . . . , N } d has coordinates α , . . . , α d . We define that a set I ⊂ { , . . . , N } is an independent set for T if for any α = ( α , . . . , α d ) ∈ I d such that c α (cid:54) = 0 we have that α = . . . = α d . N ALMOST CAP SETS IN THREE VARIABLES AND THE MULTIVARIABLE CAP SET PROBLEM 3
Theorem 2.1 (Lovett [4], Theorem 1.7) . There exists a constant c = C ( d, q ) such that for any d -tensor T we have s-rank( T ) ≥ c |I| , for any independent set I ⊂ { , . . . , N } . Proof of Theorem 1.1
Let δ > a = ( a, b, c ) ∈ ( F q \ { } ) be such that a + b + c = 0. Let (cid:15) >
0, and we assume that A ⊂ F nq is an ( (cid:15), δ ) − cap set. I.e., there exists a subset A (cid:48) ⊂ A with | A (cid:48) | ≥ δ | A | and such that for every x ∈ A (cid:48) thereare at most | A | (cid:15) pairs ( y, z ) ∈ A with ax + by + cz = 0 . Denote by F ( x, y, z ) = δ o n ( ax + by + cz ). Then we have F ( x, y, z ) = (cid:80) α ∈ A c α δ α ( x ) δ α ( y ) δ α ( z ), for c α = aα + bα + cα = 00 otherwise . Following the construction (2), the corresponding 3-tensor T F on V = { f : A → F q } is equal to T F ( f , f , f ) = (cid:88) α ∈ A c α f ( α ) f ( α ) f ( α ) . By the assumption on A (cid:48) ⊂ A , we deduce that(3) |{ c α (cid:54) = 0 | α ⊂ ( A (cid:48) ) }| ≤ δ − | A (cid:48) | ε . Using Caro-Wei lower bound [1],[6] on the independence number in a 3-uniform hypergraph, there exists
I ⊂ A (cid:48) an independent set satisfying |I| ≥ C (cid:88) x ∈ A (cid:48) d x + 1) , where d x = |{ c α (cid:54) = 0 | α = x, α ⊂ ( A (cid:48) ) }| and C > (cid:80) x ∈ A (cid:48) d x ≤ δ − | A (cid:48) | ε . Therefore,there exists a constant C > |I| ≥ C | A (cid:48) | − (cid:15)/ . Finally, using Theorem 2.1 and Lemma 2.1, there exists a constant C = C ( q, d, δ ) > F ) ≥ C | A | − (cid:15)/ . On the other hand, using the fact that F ( x, y, z ) is a polynomial in the coordinates of x, y and z , it wasproved in [3] that s-rank( F ) < b nq for a positive constant b q < q . Finally, we choose (cid:15) > b − (cid:15)/ q < q, and take any c q that satisfies b − (cid:15)/ q < c q < q . Then the statement of the Theorem holds true for chosen (cid:15) and c q . (cid:3) ALEXANDER FISH AND DIBYENDU ROY Proof of Theorem 1.2
By rearranging, if necessary, we always can assume that for every k ≤ d − a + . . . + a k (cid:54) = 0.Let us denote by b k , k = 2 , . . . , d −
2, the quantities b k = a + . . . + a k . We apply Theorem 1.1 iteratively on the equations:(4) b d − t d − + a d − x d − + a d x d = 0 , and b d − t d − + a d − x d − = b d − t d − ... b k t k + a k +1 x k +1 = b k +1 t k +1 ... b t + a x = b t and a x + a x = b t . Fix δ >
0. By Theorem 1.1 there exists (cid:15) >
0, such that for sufficiently large n there exists a set A d − ⊂ A with | A d − | ≥ (1 − δ ) | A | and such that for every t d − ∈ A d − there are at least | A | (cid:15) pairs ( x d − , x d ) ∈ A satisfying the equation (4). Applying Theorem 1.1 once again, there exists A d − ⊂ A d − with | A d − | ≥ (1 − δ ) | A d − | such that for every t d − ∈ A d − there are at least | A d − | (cid:15) pairs ( x d − , t d − ) ∈ A d − satisfyingthe equation b d − t d − + a d − x d − = b d − t d − . In such way we construct a chain of subsets A d − ⊃ A d − ⊃ . . . ⊃ A with | A k − | ≥ (1 − δ ) | A k | , for k = 2 , . . . , d −
2. Each A k satisfies that for any t k ∈ A k there exist at least | A k +1 | (cid:15) pairs ( x k +1 , t k +1 ) ∈ A k +1 satisfying the equation b k t k + a k +1 x k +1 = b k +1 t k +1 . Using Theorem 1.1 again, there exists A ⊂ A with | A | ≥ (1 − δ ) | A | and such that for every t ∈ A thereexist at least | A | (cid:15) pairs ( x , x ) ∈ A satisfying a x + a x = b t . Finally, we construct a solution for (1) consisting of distinct elements of A as follows. Take a pair of distinct( x , x ) ∈ A ⊂ A satisfying that a x + a x = b t for t ∈ A . Then there exist at least | A | (cid:15) pairs( x , t ) ∈ A ⊂ A satisfying b t + a x = b t for t that we already chosen. Find ( x , t ) among thesesolutions such that x (cid:54)∈ { x , x } . Assume that we already constructed distinct { x , . . . , x k } ∈ A k satisfyingthat a x + . . . + a k x k = b k t k , for some t k ∈ A k . Since there exist at least | A k +1 | (cid:15) pairs ( x k +1 , t k +1 ) ∈ A k +1 satisfying b k t k + a k +1 x k +1 = b k +1 t k +1 , we can choose one of the solutions ( x k +1 , t k +1 ) ∈ A k +1 satisfying that x k +1 (cid:54)∈ { x , . . . , x k } . Notice thatthere exists t k +1 ∈ A k +1 such that the sequence ( x , . . . , x k +1 ) ∈ A k +1 satisfies a x + . . . + a k +1 x k +1 = b k +1 t k +1 . N ALMOST CAP SETS IN THREE VARIABLES AND THE MULTIVARIABLE CAP SET PROBLEM 5
We continue this process till we reach distinct { x , . . . , x d − } ∈ A satisfying a x + . . . + a d − x d − = b d − t d − for some t d − ∈ A d − . Since there are at least | A | (cid:15) pairs ( x d − , x d ) ∈ A satisfying b d − t d − + a d − x d − + a d x d = 0 , we can choose the solution ( x d − , x d ) ∈ A such that x d (cid:54) = x d − and x d − , x d (cid:54)∈ { x , . . . , x d − } . This finishesthe proof of the Theorem. (cid:3) Remark.
It seems to be more natural to try to prove Theorem 1.2 using directly Tao’s reformulation [5] ofCroot-Lev-Pach-Ellenberg-Gijswijt approach together with Lovett’s lower bound (Theorem 2.1) on the slicerank of the corresponding tensor. Unfortunately, this does not work out, since the lower bound on the slicerank that we obtain is not sufficiently strong.
References [1] Caro Y.,
New Results on the Independence Number , Technical Report, Tel-Aviv University, 1979.[2] Croot, E., Lev, V. F., Pach, P.,
Progression-free sets in Z n are exponentially small. Ann. of Math. (2) 185 (2017), no. 1,331–337.[3] Ellenberg, J., Gijswijt, D.,
On large subsets of F nq with no three-term arithmetic progression. Ann. of Math. (2) 185 (2017),no. 1, 339–343.[4] Lovett, S.,
The analytic rank of tensors and its applications.
Discrete Anal. 2019, Paper No. 7, 10 pp.[5] Tao, T. ,
A symmetric formulation of the Croot-Lev-Pach-Ellenberg-Gijswijt capset bound , 2016, http://terrytao.wordpre.com/2016/05/18/a .[6] Wei V. K.,
A Lower Bound on the Stability Number of a Simple Graph , Technical memorandum, TM 81–11217–9, Belllaboratories, 1981.
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