EXPONENTIAL BOUNDS FOR THE ERDŐS-GINZBURG-ZIV CONSTANT ERIC NASLUND Abstract. The Erdős-Ginzburg-Ziv constant of an abelian group G, denoted s(G), is the smallest k N such that any sequence of length s(G) contains a zero-sum 7 ∈ 1 subsequenceoflength exp(G). Inthispaper,weusethe multi-slicerankmethodfrom 0 [12], which is a variant of Tao’s slice rank method [15], to prove that 2 s Fn 3(p 1)p!(J(p)p)n n p ≤ − a where 0.8414 < J(p) < .9183(cid:0)7 is(cid:1)a constant depending on p, equal to the constant J appearing in Ellenberg and Gijswijt’s bounds for arithmetic progression-free subsets 8 inFn [4]. We alsoimproveupona recentresultofHegedűs,andshowthatif (Z/kZ)n p 1 satisfies property D as defined in [7], then O] s((Z/kZ)n)≤(n−1)γkn,q +1 where C k 1 xq −q−1 h. γk,q = q 0 0. Ellenberg and Gijswijt’s recent bounds for capsets in Fn [4] imply that 3 s((Z/3Z)n) 2(J(3)3)n ≤ where 1 J(3) = 3 207+33√33 = 0.9183.... 8 q In this paper, for p 5 we give an unconditional exponential improvement to the upper bounds for s F≥n as n , where F = (Z/pZ) denotes the finite field with p p → ∞ p elements. Our main result is: (cid:0) (cid:1) Theorem 1. Suppose that A Fn has size ⊂ p A > 3(p 1)p!(J(p)p)n | | − where 1 1 xp J(p) = inf − . p 01 3logz Since J(3) = 0.9183..., it follows that for p 3, J(p) lies in the range ≥ 0.8414 J(p) 0.9184. ≤ ≤ The proof of theorem 1 uses the multi-slice rank method, given in [12], which is a variant of slice rank method [15]. The multi-slice rank allows us to handle the indicator tensor that results from forcing the variables to be distinct, which has large slice rank for p 5. The slice rank method was introduced by Tao [15] following the work of ≥ Ellenberg and Gijswijt [4], and the breakthrough result of Croot, Lev and Pach [3]. The slice rank method has seen numerous applications, such as Tao’s proof of Ellenberg and Gijswijt’s upper bound for progression-free sets in Fn [15, 4], new bounds for the Erdős- p Szemerédi sunflower problem [13, 6], and disproving certain conjectures concerning fast matrix multiplication [2]. We refer the reader to [2] for a more detailed discussion of the properties of the slice rank. EXPONENTIAL BOUNDS FOR THE ERDŐS-GINZBURG-ZIV CONSTANT 3 We say that a group G satisfies property D (see [7, Sec. 7]) if every sequence S of length s(G) 1 that does not contain exp(G) elements summing to 0, when viewed as − a multi-set, takes the form exp(G)−1 S = T ∪i=1 for some set T. That is, if S is a maximal sequence that does not contain exp(G) elements summing to zero, then every element in S appears exactly exp(G) 1 times. Gao and Geroldinger conjecture [7, Conj. 7.2] that (Z/kZ)n satisfies prope−rty D for everyk andn, andunder thisconjecturewecanimprove upontheboundsfortheErdős- Ginzburg-Ziv constant using the slice rank method. Recently, Hegedűs [9] showed that if Fn satisfies property D, then p 1− (p−2)2 n+1 s Fn (p 1)p(cid:18) 2p2logp(cid:19) +1, p ≤ − and we give the following im(cid:0) pr(cid:1)ovement: Theorem 2. Assume that (Z/kZ)n satisfies property D. Then s((Z/kZ)n) (n 1)γn +1 ≤ − k,q where k 1 xq γk,q = inf − x−q−k1. q 0 γn | | k,q where k 1 xq γk,q = inf − x−q−k1, q 0 2 be a prime number. For x ,...,x Fn, define(cid:0) (cid:1) 1 p ∈ p F : Fn p F p p → p by (cid:0) (cid:1) n (3.1) F (x ,...,x ) = 1 (x +x + +x )p−1 , p 1 p 1i 2i pi − ··· i=1 Y(cid:0) (cid:1) where x is understood to be the ith coordinate of the vector x Fn, and the notation ji j ∈ p Fn p is used to denote the p-fold Cartesian product of Fn with itself. Then p p (cid:0) (cid:1) 1 if x +x + +x = 0 1 2 p F (x ,...,x ) = ··· . p 1 p 0 otherwise ( This tensor does not take into account whether or not the variables are distinct, and so in particular for any x Fn ∈ p F (x,...,x) = 1. p In order to modify this tensor so that it picks up only distinct k-tuples of elements summing to zero, we use the technique in [12], and introduce an indicator which is a sum over the permutations in the symmetric group S . For every σ S , define p p ∈ (3.2) f : Xp F σ → to be the function that is 1 if (x ,...,x ) is a fixed point of σ, and 0 otherwise. The 1 p following is [12, Lemma 4]: Lemma 3. We have the identity 1 if x ,...,x are distinct 1 k sgn(σ)f (x ,...,x ) = , σ 1 k 0 otherwise ( σX∈Sk where sgn(σ) is the sign of the permutation. Proof. By definition, sgn(σ)f (x ,...,x ) = sgn(σ) σ 1 k σX∈Sk σ∈SXtab(−→x) whereStab(x) S isthestabilizerof x. Sincethestabilizerisaproductofsymmetric −→ k −→ ⊂ groups, this will be non-zero precisely when Stab(x) is trivial, and hence x ,...,x −→ 1 k must be distinct. (cid:3) Using this lemma, we give the following modification of [12, Lemma 5]: Lemma 4. Let S denote the set all elements in S given as a product of i disjoint i k k C ⊂ cycles. Define R (x ,...,x ) = sgn(σ)f (x ,...,x ). k 1 k σ 1 k σXSk ∈ σ / C ,C 1 2 ∈ EXPONENTIAL BOUNDS FOR THE ERDŐS-GINZBURG-ZIV CONSTANT 8 Then for k 3, ≥ 1 if x ,...,x are distinct 1 k ( 1)k−1(k 1)! k−1 1 if x = = x R (x ,...,x ) =  − − j=2 j 1 ··· k , k 1 k α(x1,...,xk) P if x1,...,xk take on 2 distinct values  0 otherwise   where  α(x ,...,x ) = ( 1)k−1# σ that fix (x ,...,x ) . 1 k 2 1 k − { ∈ C } Proof. If there are 3 or more distinct elements among x ,...,x then f (x ,...,x ) = 0 1 k σ 1 k for any σ , , and so the identity holds by lemma 3. When there are exactly two 2 1 ∈ C C distinct elements among x ,...,x , f (x ,...,x ) = 0 for any σ , and so 1 k σ 1 k 1 ∈ C R (x ,...,x ) = ( 1)k−1# σ that fix (x ,...,x ) k 1 k 2 1 k − { ∈ C } since sgn(σ) = ( 1)k for any σ . Lastly, when x = = x , we have that 2 1 k − ∈ C ··· R (x ,...,x )+( 1)k +( 1)k−1 = 0 k 1 k 2 1 − |C | − |C | by lemma 3 since sgnσ = ( 1)k−1 for σ . Since 1 − ∈ C k−1 1 = (k 1)! and = (k 1)! , 1 2 |C | − |C | − j j=1 X it follows that k−1 1 R (x ,...,x ) = ( 1)k−1 l 1 k − j j=2 X when x = = x . (cid:3) 1 k ··· Remark 2. Let 1 x = = x 1 k δ(x ,...,x ) = ··· . 1 k 0 otherwise ( We can calculate R (x ,...,x ) exactly as a linear combination of k! (k 1)! (k k 1 k − − − − 1)! k−1 1 delta functions, including the constant function. When k = 3 j=1 j P R (x ,x ,x ) = 1, 3 1 2 3 when k = 4 R (x ,x ,x ,x ) =1 δ(x ,x ) δ (x ,x ) δ(x ,x ) 4 1 2 3 4 1 2 2 2 3 3 4 − − − δ(x ,x ) δ(x ,x ) δ(x ,x ), 4 1 1 3 2 4 − − − and when k = 5 R (x ,x ,x ,x ,x ) =1 δ(x ,x )+2 δ(x ,x ,x ) 5 1 2 3 4 5 i j i j l − i

Exponential Bounds for the Erd\H{o}s-Ginzburg-Ziv Constant

Similar Exponential Bounds for the Erd\H{o}s-Ginzburg-Ziv Constant