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SALEM SETS IN VECTOR SPACES OVER FINITE FIELDS CHANGHAOCHEN Abstract. We prove that almost allrandom subsets of afinite vector space areweakSalemsets(smallFouriercoefficient),whichextendsaresultofHayes 7 toadifferentprobabilitymodel. 1 0 2 b 1. Introduction e F Let F denote the finite field with p element where p is prime, and Fd be the p p 2 d-dimensional vector space over this field. Let E ⊂ Fdp. We use the same notation 2 as in Babai [3], Hayes [4] to define that ] Φ(E)=max E(ξ). (1) A ξ6=0 | | C Here and in what follows, we simply write Eb(x) for the characteristic function of . E, E it’s discrete Fourier transformwhich we will define it in Section2. For ξ =0, h we mean that ξ is a non-zero vector of Fd. Applying the Plancherel identity,6 we t b p a have that for any E Fd with #E pd/2, m ⊂ p ≤ [ #E/2 Φ(E) #E. (2) 2 See Babai [3, Proposition 2.6p] for mor≤e details≤. The notation #E stands for the v cardinality of a set E. Observe that the optimal decay of E(ξ) for all ξ = 0 are 8 controlledbyO(√#E). WewriteX =O(Y)meansthatthereisapositivec6onstant 5 b C such that X CY, and X = Θ(Y) if X = O(Y) and Y = O(X). Isoevich and 9 ≤ Rudnev [6] called these sets Salem sets. To be precise we show the definition here. 7 0 Definition 1.1. [6] A subset E Fd is called a Salem set if for all non-zero ξ of 1. Fd, ⊂ p p 0 E(ξ) =O( #E). (3) 7 | | p 1 NotethatthisisafinitefieldsvbersionofSalemsetsinEuclideanspaces. Roughly : speaking, a set in Euclidean space is called a Salem set if there exist measures on v i this set, and the Fourier transform of these measures have optimal decay, see [2], X [9, Chapter 3] for more details on Salem sets in Euclidean spaces. r It is well known that the sets for which all the non-zero Fourier coefficient are a small play an important role, e.g., see [3], [9] and [11]. For some applications of Salem sets in vector spaces over finite fields, see [5], [6], [7]. In [4, Theorem 1.13] Hayes proved that almost all m-subset of Fd are (weak) p SalemsetswhichansweraquestionofBabai. Tobeprecise,letE =Eω beselected uniformly at randomfrom the collectionof all subsets of Fd which have m vectors. p Let Ω(Fd,m) denotes the probability space. p Theorem1.2(Hayes). Letε>0. Letm pd/2. ForallbutanO(p−dε)probability E Ω(Fd,m), ≤ ∈ p Φ(E)<2 2(1+ε)mlogpd =O mlogpd . (4) q (cid:16)p (cid:17) 2010 Mathematics Subject Classification. 05B25,52C99. Keywords and phrases. Finitefields,Salemsets. 1 2 CHANGHAOCHEN For convenience we call this kink of subset of Fd weak Salem set. p 1.1. Percolation on Fd. Thereisaanotherrandommodelwhichiscloselyrelated p to the randommodel Ω(Fd,m). First we show this randommodel in the following. p Let 0 < δ < 1. We choose each point of Fd with probability δ and remove it with p probability 1 δ, all choices being independent of each other. Let E = Eω be the collection−of these chosen points, and Ω = Ω(Fd,δ) be the probability space. p NotethatbothrandommodelsΩ(Fd,m)andΩ(Fd,δ)arerelatedtothewellknown p p Erdo¨s-Rényi-Gilbertrandom graph models. We note that Hayes [4] proved a similar result to Theorem 1.2 for the random model Ω(Fd,1/2). However, the martingale argument for Ω(Fd,1/2) and Ω(Fd,m) p p p of[4]do notapply easilyto the randommodelΩ(Fd,δ)for othervaluesofδ =1/2. p 6 Babai [3, Theorem 5.2] used the Chernoff bounds for the model Ω(Fd,1/2), but it p seems that the method also can not be easily extended to general δ. We note that Babai [3], Hayes [4] proved their results in general finite Abelian group, see [3], [4] formoredetails. ForthefinitevectorspaceFd (specialAbelgroup)weextendtheir p result to general δ. Theorem 1.3. Let ε > 0. Let δ (0,1). For all but an O(p−dε) probability E Ω(Fd,δ), ∈ ∈ p Φ(E)<2 (1+ε)δpdlogpd =O δpdlogpd . (5) q (cid:16)p (cid:17) We know that almost all set E Ω(Fd,δ) has size roughly δpd. This follows by ∈ p Chebyshev’s inequality, 1 4pdδ(1 δ) 1 P(#E pdδ pdδ) − =O . (6) | − |≥ 2 ≤ (pdδ)2 (cid:18)δpd(cid:19) We immediately have the following corollary, which says that almost all E Ω(Fd,δ) is a weak Salem set. ∈ p Corollary 1.4. Let ε > 0. Let δ (0,1). For all but an O(max p−dε, 1 ) ∈ { δpd} probability E Ω(Fd,δ), ∈ p E(ξ) =O #Elogpd . (7) | | (cid:16)p (cid:17) In Fd, it seems that the onbly knownexamples of Salem sets are discrete parabo- p loid anddiscrete sphere. We note that both the size of the discrete paraboloidand the discrete sphere are roughly pd−1, see [6] for more details. It is natural to ask thatdoesthereexistsSalemsetwithanygivensizem pn. Theaboveresultsand ≤ [8, Problem 20] suggest the following conjecture. Conjecture 1.5. Let s (0,d) be a non-integer and C be a positive constant. ∈ Then Φ(E) min as p , E √#E →∞ →∞ where the minimal taking over all subsets E Fd with ps/C #E Cps. ⊂ p ≤ ≤ 2. Preliminaries In this section we show the definition of the finite field Fourier transform, and some easy facts about the random model Ω(Fd,δ). Let f : Fd C be a complex p p −→ value function. Then for ξ Fd we define the Fourier transform ∈ p f(ξ)= f(x)e−2πipx·ξ, (8) xX∈Fd b p SALEM SETS IN VECTOR SPACES OVER FINITE FIELDS 3 where the inter product x ξ is defined as x ξ + +x ξ . Recall the following 1 1 p p · ··· Plancherel identity, f(ξ)2 =pd f(x)2. | | | | ξX∈Fd xX∈Fd p b p Specially for the subset of E Fd, we have ⊂ p E(ξ)2 =pd#E. (9) | | ξX∈Fd p b For more details on discrete Fourier analysis, see Stein and Shakarchi [10]. Weshowsomeeasyfactsaboutthe randommodelΩ(Fd,δ)inthe following. Let p ξ =0, then the expectation of E(ξ) is 6 E(E(bξ))=δ e−2πipx·ξ =0. xX∈Fd b p Since E(ξ)2 = E(x)E(y)e−2πi(xp−y)·ξ | | x,Xy∈Fd b p = E(x)+ E(x)E(y)e−2πi(xp−y)·ξ, xX∈Fd x6=Xy∈Fd p p we have E E(ξ)2 =δpd+δ2 e−2πi(xp−y)·ξ (cid:16)| | (cid:17) x6=Xy∈Fd b p =pdδ(1 δ). − We may read this identity as (for small δ) E(ξ) =Θ pdδ =Θ #E . | | (cid:16)p (cid:17) (cid:16)p (cid:17) b 3. Proof of Theorem 1.3 For the convenience to our use, we formulate a special large deviations estimate inthe following. Formorebackgroundanddetailsonlargedeviationsestimate,see Alon and Spencer [1, Appendix A]. Lemma3.1. Let X N bea sequenceindependent random variables with X { j}j=1 | j|≤ 1, µ := N E(X ), and µ := N E(X2). Then for any α>0, 0<λ<1, 1 j=1 i 2 j=1 j P P N P( Xj α) e−λα+λ2µ2(eλµ1 +e−λµ1). (10) ≥ ≤ (cid:12)Xj=1 (cid:12) (cid:12) (cid:12) Proof. Applying Markov’sinequalityto the randomvariableeλPNj=1Xj. This gives N P( Xj α)=P(eλPNj=1Xj >eλα) ≥ Xj=1 e−λαE(eλPNj=1Xj) (11) ≤ N =eλα E(eλXj), jY=1 the last equality holds since X is a sequence independent random variables. j j { } For any x 1 we have | |≤ ex 1+x+x2. ≤ 4 CHANGHAOCHEN Since λX 1, we have j | |≤ eλXj 1+λX +λ2X2, ≤ j j and hence E(eλXj) 1+E(λX )+E(λ2X2) ≤ i j eE(λXi)+E(λ2Xj2). ≤ Combining this with (11), we have N P( Xj α) e−λα+λµ1+λ2µ2. ≥ ≤ Xj=1 Applying the similar way to the above for P( N X α), we obtain j=1 j ≥− P N P( Xj α) e−λα−λµ1+λ2µ2. − ≥ ≤ Xj=1 Thus we finish the proof. (cid:3) The following two easy identities are also useful for us. cos2πx·ξ =Re e−2πipx·ξ=0 p xX∈Fd xX∈Fd p  p  (12) 2πx ξ 1+cos4πx·ξ 1 cos2 · = p = pd p 2 2 xX∈Fd xX∈Fd p p Proof of Theorem 1.3. Let ξ =0 and E Ω(Fd,δ). Let 6 ∈ p E(ξ)= E(x)e−2πipx·ξ = +i R I xX∈Fd b p where and is the real part of E(ξ), and is the imagine part of E(ξ). First we R I provide the estimate to the real part . By the Euler identity, we have b R b 2πx ξ = E(x)cos( · ). R p xX∈Fd p Note that 2πx ξ E(x)cos · ,x Fd (cid:18) p (cid:19) ∈ p isasequenceofindependentrandomvariables. Furthermore,applyingtheidentities (12) , we have 1 µ =0, µ = pdδ. (13) 1 2 2 Here µ ,µ are defined as the same way as in the Lemma 3.1. Let 1 2 α α:= 2(1+ε)pdδlogpd, λ:= . (14) q pdδ Note that λ 1 for large p. Applying Lemma 3.1, we have ≤ P( α) 2e−λα+λ2µ2 |R|≥ ≤ − α2 2 (15) =2e 2pdδ = . pd(1+ε) SALEM SETS IN VECTOR SPACES OVER FINITE FIELDS 5 Now we turn to the imagine part . Applying the similar argument to the real I part , note that the identities (12) also hold if we take sin instead of cos, we R obtain 2 P( α) . |I|≥ ≤ pd(1+ε) Combining this with the estimate (15), we obtain 4 P(E(ξ) √2α) P( α)+P( α) (16) | |≥ ≤ |R|≥ |I|≥ ≤ pd(1+ε) Observethatthbe aboveargumentworkstoanynon-zerovectorξ. Therefore,we obtain 4 P( ξ =0, s.t E(ξ) √2α) . (17) ∃ 6 | |≥ ≤ pdε Recall the value of α in (14), b α= 2(1+ε)pdδlogpd, q this completes the proof. (cid:3) Acknowledgements. 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[7] D.KohandChun-YenShen,AdditiveenergyandtheFalconerdistanceprobleminfinitefields, https://arxiv.org/pdf/1010.1597 [8] Hausdorffdimension,projections,andtheFouriertransform,Publ.Mat.48(2004), 3-48. [9] P.Mattila,FourieranalysisandHausdorffdimension,CambridgeStudiesinAdvancedMath- ematics,vol.150,CambridgeUniversityPress,2015. [10] E.SteinandR.Shakarchi,FourierAnalysis: AnIntroduction.PrincetonandOxford: Prince- tonUP,2003.Print.PrincetonLectures inAnalysis. [11] T.TaoandV.Vu,AdditiveCombinatorics,CambridgeUniversityPress. Schoolof MathematicsandStatistics, TheUniversity of NewSouthWales, Sydney NSW2052,Australia E-mail address: [email protected]