Invariance principle on the slice

We prove an invariance principle for functions on a slice of the Boolean cube, which is the set of all vectors {0,1}^n with Hamming weight k. Our invariance principle shows that a low-degree, low-influence function has similar distributions on the slice, on the entire Boolean cube, and on Gaussian space. Our proof relies on a combination of ideas from analysis and probability, algebra and combinatorics. Our result imply a version of majority is stablest for functions on the slice, a version of Bourgain's tail bound, and a version of the Kindler-Safra theorem. As a corollary of the Kindler-Safra theorem, we prove a stability result of Wilson's theorem for t-intersecting families of sets, improving on a result of Friedgut.

The CDF distance between the distribution of f on the slice ν pn and the distribution of f under the product measure µ p with marginals Berppq is at most ǫ: for all σ P R, | Pr νpn rf ă σs´Pr µp rf ă σs| ă ǫ.
This result is proved in Section 4.2.
Paper organization Some preliminaries are described in Section 2. We examine harmonic multilinear polynomials in Section 3. We prove the invariance principle in Section 4. Section 5 proves Majority is Stablest, and Section 6 proves Bourgain's tail bound, two applications of the main invariance principle. Section 7 deduces a version of the Kindler-Safra theorem from Bourgain's tail bound. Our stability result for t-intersecting families appears in Section 8. Some open problems are described in Section 9.

Preliminaries
Notation The notation 1 E is the characteristic function of the event E. Expectation, variance and covariance are denoted by E, V and Cov, respectively. The sign function is denoted sgn. The notation rns denotes the set t1, . . . , nu. The slice`r ns k˘c onsists of all subsets of rns of cardinality k. We often identify subsets of rns with their characteristic vectors in t0, 1u n .
The notation Binpn, pq denotes a binomial distribution with n trials and success probability p. The notation Popλq denotes a Poisson distribution with expectation λ. The notation Npµ, Σ 2 q denotes a normal distribution with mean µ and covariance matrix Σ 2 . For a scalar p, we use p to denote a constant p vector (of appropriate dimension which is clear from context) and I n to denote the nˆn identity matrix.
For a probability distribution π, }f } " }f } π " a E π rf 2 s is the L2 norm of f with respect to π. Note that }f } 1 " Er|f |s.
The notation a b denotes the falling factorial function: a b " apa´1q¨¨¨pa´b`1q. Asymptotic notation (Op¨q and the like) will always denote non-negative expressions. When the expression can be positive or negative, we use the notation˘Op¨q. The underlying limit is always n Ñ 8. If the hidden constant depends on variables V , we use the notation O V p¨q.
A C-Lipschitz functional is a function ψ : R Ñ R satisfying |ψpxq´ψpyq| ď C|x´y| which implies that for functions f, g on the same domain: Probability distributions Our argument will involve several different probability distributions on R n (where n will always be clear from context): • µ p is the product distribution supported on t0, 1u n given by µ p pSq " p |S| p1´pq n´|S| .
• ν k is the uniform distribution on the slice`r ns k˘.
Proof. Let S " X 1`¨¨¨`Xn , and consider the multivariate Gaussian distribution pX 1 , . . . , X n , Sq, whose distribution is easily calculated to be Np`p pn˘, pp1´pqˆI n 1 1 1 n˙q . Let pY 1 , . . . , Y n q be the distribution of pX 1 , . . . , X n , Sq conditioned on S " qn, which is well-known to be multivariate Gaussian. Using well-known formulas, the mean of this distribution is p`1n´1pqn´pnq " q (as can be derived directly), and its covariance matrix is pp1´pqpI n´1 n´11 1 q. The diagonal elements are VrY i s " pp1´pqp1´1 n q and the off-diagonal ones are CovpY i , Y j q " pp1´pqp´1 n q.
We can also go in the other direction.
Proof. As is well-known, Y 1 , . . . , Y n is a multivariate Gaussian, and it is easy to see that its mean is p. We have VrY i s " VrX i s`VrY s " pp1´pq and CovpY i , Y j q " CovpX i , X j q`VrY s " 0. The lemma follows.
The distributions µ p and ν k are very close for events depending on op ? nq coordinates.
This completes the proof.

Harmonic multilinear polynomials
Our argument involves extending a function over a slice`r ns k˘t o a function on R n , just as in the classical invariance principle, a function on t0, 1u n is extended to R n by writing it as a multilinear polynomial. In our case, the correct way of extending a function over a slice to R n is by interpreting it as a harmonic multilinear polynomial. Our presentation follows [Fil14], where the proofs of various results claimed in this section can be found. The basis in Definition 2.2 below also appears in earlier work of Srinivasan [Sri11], who constructed it and showed that it is orthogonal with respect to all exchangeable measures. Definition 2.1. Let f P Rrx 1 , . . . , x n s be a formal polynomial. We say that f is multilinear if B 2 f Bx 2 i " 0 for all i P rns. We say that f is harmonic if The somewhat mysterious condition of harmonicity arises naturally from the representation theory of the Johnson association scheme. Just as any function on the Boolean cube t0, 1u n can be represented uniquely as a multilinear polynomial (up to an affine transformation, this is just the Fourier-Walsh expansion), every function on the slice`r ns k˘c an be represented uniquely as a harmonic multilinear polynomial, using the identificationˆr ns k˙" tpx 1 , . . . , x n q P t0, 1u n : . Every real-valued function f on the slice`r ns k˘c an be represented uniquely as a harmonic multilinear polynomial of degree at most minpk, n´kq.
There is a non-canonical Fourier expansion defined for harmonic multilinear polynomials.
Definition 2.2. Let A " pa 1 , . . . , a d q and B " pb 1 , . . . , b d q be two sequences of some common length d of distinct elements of rns. We say that A ă B if: The collection of all top sets of length d is denoted B n,d , and the collection of all top sets is denoted B n .
Lemma 2.6 ([Fil14, Theorem 3.1,Theorem 3.2]). Let π be any exchangeable distribution on R n (that is, π is invariant under permutation of the coordinates). The collection B n forms an orthogonal basis for all harmonic multilinear polynomials in Rrx 1 , . . . , x n s (with respect to π), and and }χ B } π denotes the norm of χ B with respect to π.
In particular, if f is a harmonic multilinear polynomial then Erf s is the same under all exchangeable measures. Lemma 2.3 and Lemma 2.6 put together have the surprising consequence that harmonic multilinear functions have exactly the same distribution under G p and γ p,q . Lemma 2.7. Let f be a harmonic multilinear polynomial. The random variables f pG p q, f pγ p,q q are identically distributed.
Proof. According to Lemma 2.3, if px 1 , . . . , x n q " γ p,q , y " Npq´p, pp1´pq n q and y i " x i`y , then py 1 ,¨¨¨, y n q " G p . The lemma follows since y i´yj " x i´xj and a harmonic multilinear polynomial can be expressed as a function of the differences x i´xj for all i, j. Lemma 2.6 allows us to compare the norms of a harmonic multilinear function under various distributions.
Corollary 2.10. Suppose f is a harmonic multilinear polynomial of degree d on n variables. For any p ď 1{2 such that d ď pn and any π P tµ p , G p u we have }f } νpn " }f } πˆ1˘Oˆd 2 pp1´pqn˙˙.

Analysis of functions
We consider functions on three different kinds of domains: the Boolean cube t0, 1u n , the slice`r ns k˘, and Gaussian space R n . We can view a multilinear polynomial in Rrx 1 , . . . , x n s as a function over each of these domains in the natural way.
For each of these domains, we proceed to define certain notions and state some basic results. The material for the Boolean cube and Gaussian space is standard, and can be found for example in [O'D14].
Functions on the Boolean cube The Boolean cube is analyzed using the measure µ p for an appropriate p. The Fourier characters ω S and Fourier expansion of a function f : t0, 1u n Ñ R are given by We define f "k " ř |S|"kf pSqω S , and so a multilinear polynomial f of degree d can be decomposed as f " f "0`¨¨¨`f "d . Since the Fourier characters are orthogonal, the parts f "0 , . . . , f "d are orthogonal. In the future it will be convenient to separate f into f " f ďk`f ąk for an appropriate k, where f ďk " f "0`¨¨¨`f "k and f ąk " f "k`1`¨¨¨`f "d .
Define f ris pxq " f px ris q, where x ris results from flipping the ith coordinate of x. The ith cube-influence is given by The total influence of f is Inf c rf s " ř n i"1 Inf c i rf s, and it satisfies the Poincaré inequality Vrf s ď pp1´pq Inf c rf s ď pdeg f q Vrf s.
The noise operator T ρ is defined by The noise stability of f at ρ is The noise operator (and so noise stability) can also be defined non-spectrally. We have pT ρ f qpxq " Erf pyqs, where y is obtained from x by letting y i " x i with probability ρ, and y i " µ p otherwise.
Functions on the slice The slice`r ns k˘i s analyzed using the measure ν k . The corresponding notion of Fourier expansion was described in Section 2.1. A harmonic multilinear polynomial f of degree d can be decomposed as f " f "0`¨¨¨`f "d , where f "k contains the homogeneous degree k part. The parts f "0 , . . . , f "d are orthogonal.
The pi, jqth influence of a function f is Inf s ij rf s " Erpf´f pijq q 2 s, where f pijq pxq " f px pijq q, and x pijq is obtained from x by swapping the ith and jth coordinates. We define the ith influence by Inf s i rf s " 1 n ř n j"1 Inf s ij rf s, and the total influence by Inf s rf s " ř n i"1 Inf s i rf s. The total influence satisfies the Poincaré inequality Vrf s ď Inf s rf s ď pdeg f q Vrf s.
The noise operator H ρ is defined by The noise stability of f at ρ is The noise operator (and so noise stability) can also be defined non-spectrally. We have pH ρ f qpxq " Erf pyqs, where y is obtained from x by taking Pop n´1 2 log 1 ρ q random transpositions.
Functions on Gaussian space Gaussian space is R n under a measure G p for an appropriate p. In this paper, we mostly consider functions on R n given by multilinear polynomials, and these can be expanded in terms of the ω S . General functions can be expanded in terms of Hermite functions. Every squareintegrable function can be written as f " ř kě0 f "k , where f "k satisfies f "k pαx 1`p , . . . , αx n`p q " α k f "k px 1`p , . . . , x n`p q.
The distributions µ p and G p have the same first two moments, and this implies that E µp rf s " E Gp rf s and }f } µp " }f } Gp for every multilinear polynomial f . The Ornstein-Uhlenbeck operator U ρ is defined just like T ρ is defined for the cube. Noise stability is defined just like in the case of the cube, and we use the same notation S c for it.
Homogeneous parts For a function f , we have defined f "k in three different ways, depending on the domain. When f is a harmonic multilinear polynomial, all three definitions coincide. Indeed, any harmonic multilinear polynomial is a linear combination of functions of the form χ A,B . We show that under all three definitions. Let A " a 1 , . . . , a k and B " b 1 , . . . , b k . Since χ A,B is homogeneous of degree k as a polynomial, we see that χ A,B " χ "k A,B over the slice. Also, Opening the product into a sum of terms, we can identify each term with a basis function ω S for some S of size k. This shows that χ A,B " χ "k A,B over the cube. Finally, since χ A,B is harmonic, in order to show that χ A,B " χ "k A,B in Gaussian space, it suffices to show that χ A,B pαxq " α k χ A,B pxq, which is true since χ A,B is homogeneous of degree k as a polynomial.

Degrees
The following results state several ways in which degree for functions on the slice behaves as expected.
First, we show that degree is subadditive.
Lemma 2.11. Let f, g be harmonic multilinear polynomials, and let h be the unique harmonic multilinear polynomial agreeing with f g on the slice`r ns k˘. Then deg h ď deg f`deg g. Proof. We can assume that deg f`deg g ď k, since otherwise the result is trivial.
Let E i be the operator mapping a function φ on the slice to the function φ "i on the slice. That is, we take the harmonic multilinear representation of φ, extract the i'th homogeneous part, and interpret the result as a function on the slice. Also, let E ďd " ř d i"0 E i . A function φ on the slice has degree at most d if and only if it is in the range of E ďd .
Qiu and Zhan [QZ07] (see also Tanaka [Tan09]) show that f g is in the range of E ďdeg f˝Eďdeg g , where is the Hadamard product. The operators E i are the primitive idempotents of the Johnson association scheme (see, for example, [BI84, §3.2]). Since the Johnson association scheme is Q-polynomial (cometric), the range of E ďdeg f˝Eďdeg g equals the range of E ďdeg f`deg g , and so deg f g ď deg f`deg g.
As a corollary, we show that "harmonic projection" doesn't increase the degree.
Corollary 2.12. Let f be a multilinear polynomial, and let g be the unique harmonic multilinear polynomial agreeing with f on the slice`r ns k˘. Then deg g ď deg f .
Proof. When f " x 1 , one checks that g is given by the linear polynomial The corollary now follows from Lemma 2.11 and from the easy observation degpαF`βGq ď maxpdeg F, deg Gq.
An immediate corollary is that degree is substitution-monotone.
Corollary 2.13. Let f be a harmonic multilinear polynomial, let gpx 1 , . . . , x n q " f px 1 , . . . , x n´1 , bq for b P t0, 1u, and let h be the unique harmonic multilinear polynomial agreeing with g on the slice`r ns k˘. Then deg h ď deg f .

Noise operators
We have considered two noise operators, H ρ and T ρ " U ρ . Both can be applied syntactically on all multilinear polynomials. The following result shows that both operators behave the same from the point of view of Lipschitz functions.

On harmonicity
Let f be a function on the Boolean cube t0, 1u n , and letf be the unique harmonic function agreeing with f on the slice`r ns pn˘. We callf the harmonic projection of f with respect to the slice`r ns pn˘. In this section we prove Theorem 3.3, which shows that when f depends on p1´ǫq log n variables, it is close to its harmonic projection under the measure µ p . Together with Corollary 2.10, this allows us to deduce properties of f on the slice given properties of f on the Boolean cube, an idea formalized in Corollary 3.4.
We start by examining single monomials.
Proof. Without loss of generality we can assume that m " x n´d`1¨¨¨xn . Let B " tn´d`1, . . . , nu. Recall that the basis element χ B is equal to Let f be the unique harmonic multilinear polynomial agreeing with m on`r ns k˘. Corollary 2.12 shows that deg f ď d. The coefficientf pBq of χ B in the Fourier expansion of f is given by the formulaf pBq " xf, since there are pn´dq d summands in the definition of χ B . The value of }χ B } 2 is given by Lemma 2.6 and Lemma 2.9: We proceed to compute xf, n´2d`1 n´d`1 .
Finally, since c m ‰ 0 and deg f ď d, we can conclude that deg f " d.
As a consequence, we obtain a result on Fourier characters on the cube.
Let ω " ω S be a Fourier character with respect to the measure µ p of degree d, and letω be the unique harmonic multilinear polynomial agreeing with ω on`r ns np˘( where d ď np ď n{2). For π P tµ p , G p u we have where η involves other monomials. In fact, sinceω is harmonic, it is invariant under shifting all the variables by p, and soω " cω`η 1 , where η 1 involves other characters. Due to orthogonality of characters we have Sinceω is harmonic, Corollary 2.10 allows us to estimate }ω} 2 π given }ω} 2 νpn , which we proceed to estimate: .
Corollary 2.10 shows that the same estimate holds even with respect to π, and so We can now conclude that a multilinear polynomial depending on a small number of variables is close to its harmonic projection.
Proof. We can assume without loss of generality that f depends on the first d coordinates. Express f as a linear combination of characters: f " ř SĎrdsf pSqω S . Clearlyf " ř SĎrdsf pSqω S , whereω S is the unique function agreeing with ω S on`r ns pn˘. Lemma 3.2 together with the Cauchy-Schwartz inequality shows that This completes the proof.
Combining Theorem 3.3 with Corollary 2.10, we show how to deduce properties of f on the slice given its properties on the cube. Corollary 3.4. Let f be a multilinear polynomial depending on d variables, and letf be the unique harmonic multilinear polynomial agreeing with f on`r ns pn˘, where d ď pn ď n{2. Suppose that }f } 2 µp " }f } 2 Gp ď 1. For π P tµ p , G p u we have: Proof. Throughout the proof, we are using Corollary 2.10 to convert information onf with respect to π to information onf with respect to ν pn . All calculations below are with respect to π.
For the first item, note that The second item follows from the triangle inequality For the third item, notice first that | Erf s| ď }f } 1 ď }f } 2 " 1. The item now follows from the previous two.
The fourth item follows from the fact that S c ρ is 1-Lipschitz, which in turn follows from the fact that S c ρ rf s " }T ? ρ f } 2 and that T ? ρ is a contraction. For the fifth item, assume that f depends on the first d variables, and write f " ř SĎrds c S ω S . We havẽ The fifth item now follows from the triangle inequality and the second item.

Invariance principle
In the sequel, we assume that parameters p P p0, 1{2s and n such that pn is an integer are given. The assumption p ď 1{2 is without loss of generality. We will use big O notation in the following way: f " O p pgq if for all n ě N ppq, it holds that f ď Cppqg, where N ppq, Cppq are continuous in p. In particular, for any choice of p L , p H satisfying 0 ă p L ď p H ă 1, if p P rp L , p H s then f " Opgq. Stated differently, as long as λ ď p ď 1´λ, we have a uniform estimate f " O λ pgq. Similarly, all constants depending on p (they will be of the form A p for various letters A) depend continuously on p.
Proof sketch Let ψ be a Lipschitz functional and f a harmonic multilinear polynomial of unit variance, low slice-influences, and low degree d. A simple argument shows that f also has low cube-influences, and this implies that The idea is now to apply the multidimensional invariance principle jointly to f and to S " x1`¨¨¨`xn´np ?  Since Pr Gp r|S| ď σs « Pr µp r|S| ď σs " Θ p pσq, we can conclude that By choosing σ appropriately, we balance the two errors and obtain our invariance principle. For minor technical reasons, instead of using 1 |S|ďσ we actually use a Lipschitz function supported on |S| ď σ.
Main theorems Our main theorem is Theorem 4.6, proved in Section 4.1 on page 16. This is an invariance principle for low-degree, low-influence functions and Lipschitz functionals, comparing the uniform measure on the slice ν pn to the measure µ p on the Boolean cube and to the Gaussian measure G p .
Some corollaries appear in Section 4.2 on page 17. Corollary 4.7 gives a bound on the Lévy distance between the distributions f pν pn q and f pG p q for low-degree, low-influences functions. Corollary 4.9 gives a bound on the CDF distance between the distributions f pν pn q and f pG p q for low-degree, low-influences functions. Corollary 4.10 extends the invariance principle to functions of arbitrary degree to which a small amount of noise has been applied.

Main argument
We start by showing that from the point of view of L2 quantities, distributions similar to µ p behave similarly.
Definition 4.1. Let p P p0, 1q. A parameter q P p0, 1q is p-like if |p´q| ď a pp1´pq{n. A distribution is p-like if it is one of the following: µ q , ν qn , G q , where q is p-like. n, and let π 1 , π 2 be two p-like distributions. Then The same holds if we replace }f } 2 with Inf s ij rf s " }f´f pijq } 2 or Inf c i rf s " } Bf Bxi } 2 . Furthermore, there is a constant S p such that if d ď S p ? n then for all p-like distributions π 1 , π 2 , Proof. Let α D pqq " p2qp1´qqq D , where D ď d. An easy calculation shows that α 1 D pqq " 2p1´2qqDp2qp1q qq D´1 , and in particular |α 1 D pqq| " OpDα D pqq{qp1´qqq. It follows that for p-like q, α D pqq " α D ppqp1Ȏ p pD{ ? nqq. Lemma 2.9 thus shows that for π P tµ q , ν qn , G q u and all D ď d, Since D ď d and d ď ?
n implies d 2 {n ď d{ ? n, we conclude that The lemma now follows from Corollary 2.8.
We single out polynomials whose degree satisfies d ď S p ? n.
Definition 4.2. A polynomial has low degree if its degree is at most S p ? n, where S p is the constant in Lemma 4.1.
We can bound the cube-influence of a harmonic multilinear polynomial in terms of its slice-influence.
Lemma 4.2. Let f be a harmonic multilinear polynomial of low degree d, and let π be a p-like distribution. For all i P rns, with respect to π: Proof. We will show that for the product measure π " µ p it holds that Vrf s`2 n pp1´pqpn´dq Inf s i rf s which will imply the statement of the lemma by Lemma 4.1.
The idea is to come up with an explicit expression for Inf s i rf s.
On the other hand, we have The L2 triangle inequality shows thatf pS Y tiuq 2 ď 2f pS Y tjuq 2`2 pf pS Y tiuq´f pS Y tjuqq 2 , and so Using Lemma 4.2, we can show that the behavior of a low degree function isn't too sensitive to the value of q in ν qn . Lemma 4.3. Let f be a harmonic multilinear polynomial of low degree d, and let ℓ be an integer such that ν ℓ is p-like. For every C-Lipschitz functional ψ, Proof. Let q " ℓ{n, which is p-like. For i P rns, let pX i , Y i q be the distribution obtained by choosing a random X i P`r nsztiu ℓ˘a nd setting Y i " X i Y tiu. Note that f pX i q´f pY i q " pf´f ris qpX i q. Since Pr ν ℓ rx i " 0s " 1´q, we have Erpf pX i q´f pY i qq 2 s ď p1´qq´1 Inf c i rf s ν ℓ " O pˆd n Vrf s νpn`I nf s i rf s νpn˙, using Lemma 4.2 and Lemma 4.1.
Consider now the distribution pX, Y q supported on`r ns ℓ˘ˆ`r ns ℓ`1˘o btained by taking X " ν ℓ and choosing Y Ą X uniformly among the n´ℓ choices; note that Y " ν ℓ`1 . Since pX, Y q is a uniform mixture of the distributions pX i , Y i q, we deduce  . Let Q 1 , . . . , Q k be n-variate multilinear polynomials of degree at most d such that with respect to µ p , VrF i s ď 1 and Inf c j rF i s ď τ for all i P rks and j P rns. For any C-Lipschitz functional Ψ : R k Ñ R (i.e., a function satisfying |Ψpxq´Ψpyq| ď C}x´y} 2 ), for some (explicit) constant ρ p ě 1. Let f be a harmonic multilinear polynomial of low degree d ě 1 such that with respect to µ p , Erf s " 0, Vrf s ď 1 and Inf s i rf s ď τ for all i P rns. Suppose that τ ď R´d p and n ě R d p , for some constant R p . For any C-Lipschitz functional ψ such that ψp0q " 0 and B-Lipschitz functional φ (where B ě 1) satisfying }φ} 8 ď 1, The condition Inf s i rf s ď τ for all i P rns can be replaced by the condition Inf c i rf s µp ď τ for all i P rns. Proof. For M to be chosen later, definẽ ψpxq " It is not hard to check thatψ is also C-Lipschitz.
We are going to apply Proposition 4.4 with Q 1 " f , Q 2 " S{ a pp1´pqn, and Ψpy 1 , y 2 q "ψpy 1 qφpy 2 q. With respect to µ p , VrQ 2 s " 1 and Inf c i rQ 2 s " 1{ppp1´pqnq for all i P rns. Lemma 4.2 shows that Inf c i rf s " O p p d n`τ q, and so the cube-influences of Q 1 , Q 2 are bounded by O p pτ`d n q. Since |Ψpy 1 , y 2 q´Ψpz 1 , z 2 q| ď |Ψpy 1 , y 2 q´Ψpy 1 , z 2 q|`|Ψpy 1 , z 2 q´Ψpz 1 , z 2 q| ď M B|y 2´z2 |`C|y 1´z1 |, we see that Ψ is pM B`Cq-Lipschitz. Therefore Next, we want to replaceψ with ψ. For π P tµ p , G p u we have {6 completes the proof. The conditions on τ, n guarantee that ρ d p pτ`d n q 1{6 ď 1, and so B ě 1 allows us to obtain the stated error bound.
In order to finish the proof, we combine Lemma 4.5 with Lemma 4.3.
Theorem 4.6. Let f be a harmonic multilinear polynomial of degree d such that with respect to ν pn , Vrf s ď 1 and Inf s i rf s ď τ for all i P rns. Suppose that τ ď I´d p δ K and n ě I d p {δ K , for some constants I p , K. For any C-Lipschitz functional ψ and for π P tG p , µ p u, The condition Inf s i rf s ď τ for all i P rns can be replaced by the condition Inf c i rf s µp ď τ for all i P rns.
Proof. We prove the theorem for π " G p . The version for µ p then follows from the classical invariance principle, using Lemma 4.2.
Replacing f with f´Erf s (recall that the expectation of f is the same with respect to both µ p and π) doesn't change the variance and influences of f , so we can assume without loss of generality that Erf s " 0. Similarly, we can replace ψ with ψ´ψp0q without affecting the quantity E νpn rψpf qs´E µp rψpf qs, and so we can assume without loss of generality that ψp0q " 0.
For a parameter σ ď 1 to be chosen later, define a function φ supported on r´σ, σs by φpxq " Note that }φ} 8 " 1 and that φ is p1{σq-Lipschitz. Lemma 4.5 (together with Lemma 4.1) shows that assuming τ ď R´d p and n ě R d p (the condition on n implies that d is low degree). Let α be the distribution of x 1`¨¨¨`xn under G p . Lemma  Therefore

Proposition 4.4 shows that
Moreover, E Gp rψpf qs ď C E Gp r|f |s " O p pCq. It follows that It is not hard to check that E Gp rφpSqs " Θ p pσq, and so for n ě A p σ´9 we have E µp rφpSqs " Θ p pσq, implying It is not hard to check that if d ď B p n β and n ě M p then n ě A p σ´9, and that if τ, n´1 ď ρ´γ d p then σ ď 1; these are the conditions necessary for our estimate to hold. In fact, for an appropriate choice of γ p ě γ, the condition n ě ρ γpd p implies the condition d ď B p n β , and furthermore allows us to estimate n´1 {60 d 3{10 " O p pn´1 {70 q (say), and to control the other error term similarly. This completes the proof of the theorem.

Corollaries
Theorem 4.6 allows us to bound the Lévy distance between the distribution of a low degree polynomial with respect to ν pn and the distribution of the same polynomial with respect to G p or µ p . This is the analog of [MOO10,Theorem 3.19].
Corollary 4.7. Let f be a harmonic multilinear polynomial of degree d such that with respect to ν pn , Vrf s ď 1 and Inf s i rf s ď τ for all i P rns. There are parameters X p , X such that for any 0 ă ǫ ă 1{2, if τ ď X´d p ǫ X and n ě X d p {ǫ X then the Lévy distance between f pν pn q and f pπq is at most ǫ, for π P tG p , µ p u. In other words, for all σ, Pr νpn rf ď σ´ǫs´ǫ ď Pr π rf ď σs ď Pr νpn rf ď σ`ǫs`ǫ.
We can similarly get a bound in the other direction. To complete the proof, choose δ " c p ǫ 2 for an appropriate c p .
Using the Carbery-Wright theorem, we can bound the actual CDF distance. This is the analog of [MOO10, Theorem 3.19(30)]. Corollary 4.9. Let f be a harmonic multilinear polynomial of degree d such that with respect to ν pn , Vrf s " 1 and Inf s i rf s ď τ for all i P rns. There are parameters Y p , Y such that for any 0 ă ǫ ă 1{2, if τ ď pY p dq´dǫ Y d and n ě pY p dq d {ǫ Y d then the CDF distance between f pν pn q and f pπq is at most ǫ, for π P tG p , µ p u. In other words, for all σ, | Pr νpn rf ď σs´Pr π rf ď σs| ď ǫ.
Proof. It is enough to prove the corollary for π " G p , the other case following from the corresponding result in the classical setting. Corollary 4.7 and the Carbery-Wright theorem show that for τ ď X´d p η X and n ě X d p {η X we have Pr νpn rf ď σs ď Pr Gp rf ď σ`ηs`η ď Pr Gp rf ď σs`O p pdη 1{d q.
We can similarly obtain a bound from the other direction. To complete the proof, choose η " c p pǫ{dq d for an appropriate c p .
All bounds we have considered so far apply only to low degree functions. We can get around this restriction by applying a small amount of noise to the functions before applying the invariance principle itself. This is the analog of [MOO10, Theorem 3.20] Even though the natural noise operator to apply on the slice is H ρ , from the point of view of applications it is more natural to use U ρ (which we apply syntactically). Lemma 2.14 shows that the difference between the two noise operators is small. Corollary 4.10. Let f be a harmonic multilinear polynomial such that with respect to ν pn , Vrf s ď 1 and Inf s i rf s ď τ for all i P rns. There is a parameter Z p such that for any 0 ă ǫ ă 1{2 and 0 ă δ ă 1{2, if τ ď ǫ Zp{δ and n ě 1{ǫ Zp{δ then for π P tG p , µ p u, Proof. Let g " U 1´δ f . Let d be a low degree to be decided, and split g " g ďd`gąd . With respect to ν pn , }g ąd } 2 " ř tąd p1´δq 2t }f "t } 2 ď p1´δq 2d . On the other hand, Theorem 4.6 shows that if τ ď I´d p ǫ K and n ě I d p {ǫ K then | E νpn rψpg ďd qs´E π rψpg ďd qs| " O p pCǫq.
Since }g´g ďd } " }g ąd }, Lemma 2.1 and Lemma 4.1 show that as long as the degree d is low, Choosing d " logp1{ǫq{δ, the resulting error is O p pCǫq. This degree is low if logp1{ǫq{δ ď S p ? n, a condition which is implied by the stated condition on n.
Therefore Borell's theorem for f and G p follows from the theorem for g and Np0, 1q.
Majority is stablest states that a similar bound essentially holds for all low influence functions on the slice. This result was originally proved using the invariance principle in [MOO10]. An alternative inductive proof appears in [DMN13].
It is known (see for example [MOO10]) that the bound Φ´1pµq is achieved by threshold functions. Corollary 2.10 together with Lemma 2.14 shows that threshold functions achieve the bound also on the slice. Indeed, take a threshold function f on d variables such that with respect to µ p , Erf s " µ and S c ρ rf s ě Γ ρ pµq´ǫ. Letf be the restriction of f to the slice`r ns pn˘. Corollary 2.10 shows that Erf s " µ˘o n p1q and S c ρ rf s " S c ρ rf s˘o n p1q. Lemma 2.14 shows that S s ρ rf s " S c ρ rf s˘o n p1q. Therefore for large n, Erf s « µ and S c ρ rf s ě Γ ρ pµq´2ǫ. The condition Inf s i rf s ď τ for all i P rns can be replaced by the condition Inf c i rf s µp ď τ for all i P rns.
Proof. We identify f with the unique harmonic multilinear polynomial agreeing with it on`r ns pn˘. For a parameter 0 ă δ ă 1{2 to be chosen later, let g " H 1´δ f . Note that the range of g on`r ns pn˘i s included in r0, 1s as well, since H 1´δ is an averaging operator. We have S s ρ rf s´S s ρ rgs " S s ρ rf s´S s ρp1´δq 2 rf s " Since d ď n{2, we have ρ dp1´pd´1q{nq p1´p1´δq 2dp1´pd´1q{nq q ď ρ d{2 p1´p1´δq 2d q ď 2δdρ d{2 .
The expansion x{p1´xq 2 " ř d dx d shows that dρ d{2 ď ? ρ{p1´?ρq 2 , and so | S c ρ rf s´S c ρ rgs| ď 2δ ? ρ p1´?ρq 2 . (1) From now on we concentrate on estimating S c ρ rgs. Define the clumped square function Sq by Sqpxq " It is not difficult to check that Sq is 2-Lipschitz. Corollary 4.10 together with Lemma 2.14 shows that for all ǫ ą 0, if τ, 1 n ď ǫ Zp{δ then We would like to apply Borell's theorem in order to bound S c ρ rgs, but g is not necessarily bounded by r0, 1s on R n . In order to handle this, we define the functiong " maxp0, minp1, gqq, which is bounded by r0, 1s. Let dist r0,1s be the function which measures the distance of a point x to the interval r0, 1s. The function dist r0,1s is clearly 1-Lipschitz, and so Corollary 4.10 implies that under the stated assumptions on τ, 1 n , we have E Gp r|g´g|s " E Gp rdist r0,1s pgqs " | E νpn rdist r0,1s pgqs´E Gp rdist r0,1s pgqs| " O p pǫq.
Theorem 6.1 (Bourgain). Let f : R n Ñ t˘1u. For any k ě 1 we have, with respect to Gaussian measure Np0, 1q, While the theorem is stated for Np0, 1q, it holds for G p as well. Indeed, given a function f , define a new function g by gpxq " f p a pp1´pqx`pq. If x " Np0, 1q then a pp1´pqx`p " G p , and so Vrf s Gp " Vrgs Np0,1q . Our definition of f "i for G p makes it clear that g "i pxq " f "i p a pp1´pqx`pq, where g "i is the degree i homogeneous part of g. This implies that }f ąk } 2 Gp " }g ąk } 2 Np0,1q . Therefore Bourgain's theorem for f and G p follows from the theorem for g and Np0, 1q.
Following closely the proof of [KKO14, Theorem 3.1], we can prove a similar result for the slice.
Theorem 6.2. Fix k ě 2. Let f :`r ns pn˘Ñ t˘1u satisfy Inf s i rf ďk s ď τ for all i P rns. For some constants W p,k , C, if τ ď W´1 p,k Vrf s C and n ě W p,k { Vrf s C then The condition Inf s i rf s ď τ for all i P rns can be replaced by the condition Inf c i rf s µp ď τ for all i P rns. Proof. We treat f as a harmonic multilinear polynomial. Since f is Boolean, working over ν pn we have }f ąk } 2 " }f´f ďk } 2 ě }f ďk´s gnpf ďk q} 2 " }f ďk } 2`1´2 Er|f ďk |s.
We do not attempt to match here [KKO14,Theorem 3.2], which has the best constant in front of Vrf s{ ? k.
7 Kindler-Safra theorem Theorem 6.2 implies a version of the Kindler-Safra theorem [KS04,Kin02], Theorem 7.5 below. We start by proving a structure theorem for almost degree k functions. We start with a hypercontractive estimate due to Lee and Yau [LY98] (see for example [Fil14,Proposition 6.2]).
Proposition 7.1. For every p there exists a constant r p such that for all functions f :`r ns pn˘Ñ R, }H rp f } 2 ď }f } 4{3 .
This implies the following dichotomy result.
Lemma 7.2. Fix parameters p and k, and let f :`r ns pn˘Ñ t˘1u satisfy }f ąk } 2 " ǫ. For any i, j P rns, either Inf s ij rf s ď ǫ{2 or Inf s i,j rf s ě J p,k , for some constant J p,k .
Since g " pf´f pijq q{2, we can bound }g ąk } 2 ď }f ąk } 2 " ǫ. Therefore p4 Inf s ij rf sq 3{2 ě r k p}g} 2´ǫ q " r k p4 Inf s ij rf s´ǫq.  Proof. Apply Lemma 7.3 with τ " J p,k to obtain a set J of size Opk{J p,k q. Lemma 7.2 with ǫ " 0 shows that for i, j R J we have Inf s ij rf s " 0, and so f is invariant under permutations of coordinates outside of J.
Using Bourgain's tail bound, we can deduce a stability version of Corollary 7.4, namely a Kindler-Safra theorem for the slice.
Theorem 7.5. Fix the parameter k ě 2. Let f :`r ns pn˘Ñ t˘1u satisfy }f ąk } 2 " ǫ. There exists a function h :`r ns pn˘Ñ t˘1u of degree k depending on O k,p p1q coordinates (that is, invariant under permutations of all other coordinates) such that for some constant C.
Proof. Let F " f ďk . We can assume that 2ǫ ă J p,k {2, since otherwise the theorem is trivial. Apply Lemma 7.3 to F with parameter τ " J p,k´2 ǫ ą J p,k {2, obtaining a set J of size Opk{τ q " O p,k p1q. It is not hard to check that Inf s ij rF s ď Inf s ij rf s ď Inf s ij rF s`2}f ąk } 2 " Inf s ij rF s`2ǫ.
Therefore if i, j R J then Inf s ij rf s ă τ`2ǫ " J k,p , and so Lemma 7.2 shows that Inf s ij rF s ď Inf s ij rf s " Opǫq. For x P t0, 1u J , let G x and g x result from F and f (respectively) by restricting the coordinates in J to the value x. It is not hard to check that Pr S"νpn rS| J " xs ě pp´O p p|J|{nqq |J| " Ω p,k p1q, as long as n ě N p,k for some constant N p,k ; if n ď N p,k then the theorem is trivial. We conclude that Inf s ij rG x s " O p,k pǫq for all i, j R J and }G x´gx } 2 " }g ąk x } 2 " O p,k pǫq. Together these imply that Inf s ij rg x s " O p,k pǫq for all i, j R J, and so Inf s i rg x s " O p,k pǫq for all i R J. We can assume that n´|J| ě n{2 (otherwise the theorem is trivial) and that the skew p x of the slice on which G x , g x are defined satisfies p x " p˘O p p|J|{nq " Θppq, and so Theorem 6.2 implies that Define a function g by gpSq " Erg S|J s. The bound on Vrg x s implies If we let h " sgn g then we obtain the desired bound }f´h} 2 ď 4}f´g} 2 .
It remains to show that h has degree k if ǫ is small enough and n is large enough. We can assume without loss of generality that J " rM s, where M is the bound on |J|. We have }f´h} 2 ě }f ąk´hąk } 2 ě p}h ąk }´?ǫq 2 . Therefore On the other hand, we can write h as a Boolean function H of x 1 , . . . , x M . Lemma 3.1 shows that deg h ď deg H, and so deg h ą k implies that deg H ą k. Corollary 3.4 (5) implies that for large enough n, }h ąk } " Ω p,H p1q. Since there are only finitely many Boolean functions on x 1 , . . . , x M which can play the role of H, we conclude that if ǫ is small enough and n is large enough then deg h ď k.
We conjecture that Theorem 7.5 holds with an error bound of O p,k pǫq rather than O p,k pǫ 1{C`1 {n 1{C q.

t-Intersecting families
As an application of Theorem 7.5, we prove a stability result for the t-intersecting Erdős-Ko-Rado theorem, along the lines of Friedgut [Fri08]. We start by stating the t-intersecting Erdős-Ko-Rado theorem, which was first proved by Wilson [Wil84].
). Let t ě 1, k ě t, and n ě pt`1qpk´t`1q. Suppose that the family F Ď`r ns k˘i s t-intersecting: every two sets in F have at least t points in common. Then: (a) |F | ď`n´t k´t˘.
(b) If n ą pt`1qpk´t`1q and |F | "`n´t k´t˘t hen F is a t-star: a family of the form F " tA Pˆr ns k˙: S Ď Au, |S| " t.
The case t " 1 is the original Erdős-Ko-Rado theorem [EKR61]. Ahlswede and Khachatrian [AK97,AK99] found the optimal t-intersecting families for all values of n, k, t.
A stability version of Theorem 8.1 would state that if |F | «`n´t k´t˘t hen F is close to a t-star. Frankl [Fra87] proved an optimal such result for the case t " 1. Friedgut [Fri08] proved a stability result for all t assuming that k{n is bounded away from 1{pt`1q.
Careful inspection of Friedgut's proof shows that it is meaningful even for sub-constant ζ, but only as long as ζ " ωp1{ ? nq. We prove a stability version of Theorem 8.1 which works all the way up to ζ " 0.
Theorem 8.3. Let t ě 2, k ě t`1 and n " pt`1qpk´t`1q`r, where r ą 0. Suppose that k{n ě λ for some λ ą 0. Suppose F Ď`r ns k˘i s a t-intersecting family of measure |F | "`n´t k´t˘´ǫ`n k˘. Then there exists a family G which is a t-star or a pt, 1q-Frankl family such that for some constant C. Furthermore, there is a constant A t,λ such that ǫ ď A t,λ minpr{k, 1q C`1 implies that G is a t-star.
We do not know whether the error bound we obtain is optimal. We conjecture that Theorem 8.3 should hold with an error bound of O t,λ pmaxpk{r, 1qǫq.
(b) If p ă 1{pt`1q and µ p pF q " p t then F is a t-star [Fri08].
(c) If t ě 2, p " 1{pt`1q and µ p pF q " p t then F is either a t-star or a pt, 1q-Frankl family [Fil13].
Friedgut [Fri08] deduces his stability version of Theorem 8.1 from a stability version of Theorem 8.4. While Friedgut's stability version of Theorem 8.4 is meaningful for all p ă 1{pt`1q, his stability version of Theorem 8.1 is meaningful only for k{n ă 1{pt`1q´ωp1{ ? nq. A more recent stability result for compressed cross-t-intersecting families due to Frankl, Lee, Siggers and Tokushige [FLST14], using completely different techniques, also requires k{n to be bounded away from 1{pt`1q.
Friedgut's argument combines a spectral approach essentially due to Lovász [Lov79] with the Kindler-Safra theorem [KS04,Kin02]. Using Theorem 7.5 instead of the Kindler-Safra theorem, we are able to obtain a stability result for the entire range of parameters of Theorem 8.1. We restrict ourselves to the case t ě 2.
Our starting point is a calculation due to Wilson [Wil84].
Theorem 8.5 ([Wil84]). Let t ě 2, k ě t`1, and n ě pt`1qpk´t`1q. There exists an`r ns k˘ˆ`r ns ks ymmetric matrix A such that A SS " 1 for all S P`r ns k˘, A ST " 0 for all S ‰ T P`r ns k˘s atisfying |S X T | ě t, and for all functions f :`r ns k˘Ñ R, Af " k ÿ e"0 λ e f "e , λ e " 1`p´1q t´1´e t´1 ÿ i"0 p´1q iˆk´1´i k´t˙ˆk´e i˙ˆn´k´e`i k´e˙ˆn´k´t`i k´t˙´1 .
The eigenvalues λ e satisfy the following properties: (a) λ 0 "`n k˘`n´t k´t˘´1 .
We need to know exact asymptotics of λ t`2 .
In particular, we can find some constant C t such that λ ě pt`1`ρq t pt`ρq t`1ˆρ´C t k˙.
Therefore for 2C t {k ď ρ ď 1, we have λ " Ω t pρq. Since λ is an increasing function of ρ, this shows that for ρ ě 2C t {k, we have λ " Ω t pminpρ, 1qq. In order to finish the proof, we handle the case ρ ď C t {k. Consider n " pt`1qpk´t`1q`1. The value of 1´λ in this case is The value of the last expression without the correction term 1´i`2 tpk´t`1q´2t`2`i is exactly 1 by Theorem 8.5, and so λ ě 2 tpk´t`1q´2t`2 " Ω tˆ1 k˙.
Since λ is increasing in ρ, this shows that for all ρ ą 0 we have λ " Ω t p1{kq. If also ρ ď C t {k then this implies that λ " Ω t pρq, finishing the proof.
We need a similar result comparing the measures of t-stars and pt, 1q-Frankl families.
As ρ Ñ 8, the lower bound tends to 1{2, and in particular, we can find c t such that for ρ ě c t we have pm´m 1 q{m 1 ě 1{3. When ρ ď c t , we clearly have pm´m 1 q{m 1 " Ω t pρq, completing the proof.
The method of Lovász [Lov79] as refined by Friedgut [Fri08] allows us to deduce an upper bound on }f ąt } 2 for the characteristic function of a t-intersecting family.
Lemma 8.8. Let t ě 2, k ě t`1 and n " pt`1qpk´t`1q`r, where r ą 0. Let F Ď`r ns k˘b e a t-intersecting family, and f its characteristic function. Then }f ąt } 2 " Oˆmaxˆk r , 1˙˙¨pm´Erf sq, where m "`n´t k´tn k˘.
This already implies that Erf s ď λ´1 0 " m. Since λ 0 " m´1 and Erf s ď m, we conclude that Lemma 8.6 completes the proof.
In order to prove our stability result, we need a result on cross-intersecting families.
Theorem 8.9 ( [FT92]). Let F Ď`r ns a˘a nd G Ď`r ns b˘b e cross-intersecting families: every set in F intersects every set in G. If n ě a`b and b ě a then |F |`|G| ďˆn b˙´ˆn´a b˙`1 ďˆn b˙.
We can now prove our stability result.
Proof of Theorem 8. 3. In what follows, all big O notations depend on t and λ. We can assume that n is large enough (as a function of t and λ), since otherwise the theorem is trivial. We use the parameter p " pk´t`1q{n which satisfies λ{2 ă p ă 1{pt`1q. Let f be the characteristic function of F , so that Erf s " m´ǫ, where m "`n´t k´t˘{`n k˘. Lemma 8.8 shows that }f ąt } 2 " Opmaxpk{r, 1qqǫ, and so Theorem 7.5 shows that }f´g} 2 ď δ for the characteristic function g of some family G depending on J " J t coordinates, for some constant J t , where δ " Opmaxppk{rq 1{C , 1qǫ 1{C1 {n 1{C q; here we use the fact that λ ď k{n ď 1{2. We want to show that if δ is small enough (as a function of t) then G must be a t-star or a pt, 1q-Frankl family; if δ is large then the theorem becomes trivial.
We start by showing that if δ is small enough then G must be t-intersecting. Suppose without loss of generality that G depends only on the first J coordinates. We will show that J " G| rJs Ď t0, 1u J must be t-intersecting. If J is not t-intersecting, then pick A, B P J which are not t-intersecting, with |A| ě |B|. Let A " tS P`r nszrJs k´|A|˘: A Y S P F u and B " tS P`r nszrJs k´|B|˘: B Y S P F u. Since n ě pt`1qk´pt 2´1 q and k ě λn, if n is large enough then pk´|A|q`pk´|B|q ď n´2J, and so Theorem 8.9 shows that |A|`|B| ď`n´J k´|B|˘. Therefore }f´g} 2 " |F △G| n k˘ě`n´J k´|A|n k˘" p |A| p1´pq J´|A|ˆ1˘Oˆ1 pp1´pqn˙˙" Ωp1q,