On uncertainty inequalities related to subcube partitions and additive energy
aa r X i v : . [ c s . D M ] S e p ON UNCERTAINTY INEQUALITIES RELATED TO SUBCUBEPARTITIONS AND ADDITIVE ENERGY
NORBERT HEGYV ´ARI
Abstract.
The additive energy plays a central role in combinatorial number the-ory. We show an uncertainty inequality which indicates how the additive energyof support of a Boolean function, its degree and subcube partition are related.AMS 2010 Primary 11B30, 11L03, Secondary 11B75Keywords: Boolean cube, Fourier analysis, Additive Combinatorics Introduction and Motivation
In the graph theory it is a well-known result among many others, that a givengraph ( V ( G ) , e ( G )) the cardinality of the maximum independent set α ( G ) and themaximum degree d fulfils the relation α ( G ) · ( d + 1) ≥ | V ( G ) | . This relation tellsus that the maximum independent set and the maximum degree can not be smallsimultaneously. In mathematics there are examples like this where there is a boundof quantities. These types of phenomenons are said to be commonly uncertaintyinequalities .In this paper we are looking for connections between parameters of Boolean func-tions and some parameters from the additive combinatorics.A Boolean function is defined as a map f : { , } n
7→ { , } , other times it is used f : {− , } n
7→ {− , } e.t.c (see [D]). We will consider the set { , } n as F n withthe usual addition on field. We convert all results to f : F n
7→ { , } which is whatwill be used. One can consider a Boolean function f as an indicator of the set A = f − (1); i.e. f ( x ) = x ∈ A x / ∈ A The influence of coordinate i on f is defined as Inf i ( f ) = P r x ∈{ , } n [ f ( x ) = f ( x + e i )],where x is uniformly distributed over { , } n , and f ( x + e i ) means that we changethe i th coordinate to 1 if x i = 0 and to 0 if x i = 1 respectively. The total influenceof f is defined to be I ( f ) := P i Inf i ( f ).For a set A ⊆ F n (and this notion is defined in all semigroups in a similar way),the additive energy of A is defined as the number of quadruples ( a , a , a , a ) forwhich a + a = a + a , formally E ( A ) := |{ ( a , a , a , a ) ∈ A : a + a = a + a }| . Clearly | A | ≪ E ( A ) ≪ | A | holds, since the quadruple ( a , a , a , a ) is always asolution and given a , a , a the term a is uniquely determined by them. This notionis introduced by Terence Tao, and plays a central role in additive combinatorics. (seee.g. [TV]).Let h ( x ) be the binary entropy function defined by h ( x ) = − x log x − (1 − x ) log(1 − x ).A decision tree which computes a Boolean function f determines a partition ofthe cube { , } n , where for every element of a given part, the value of f at each leafis the same.The subcube of { , } n is a set of vectors in the form: C = { ( ∗ , ∗ , . . . , x i , ∗ , . . . , x i , . . . x ik . . . , ∗ ) : ∗ ∈ { , }} , i.e. those vectors of { , } n in which there are k fix coordinates ( x i , . . . , x i , . . . x ik ),and the rest are free. The dimension of this subcube is 2 n − k . N UNCERTAINTY INEQUALITIES RELATED TO SUBCUBE PARTITIONS AND ADDITIVE ENERGY3
Clearly there is a partition of { , } n into the union of subcubes ∪ i C i , such thatthe value of the function f is the same on each vector of C i , i.e. for every i and x, y ∈ C i , f ( x ) = f ( y ).For example when f is a dictator function, i.e. f ( x , x , . . . , x n ) = x i for some1 ≤ i ≤ n there are at most two subcubes.However there exist a monochromatic subcube partition of { , } n which does notinduce any decision tree; one of the simplest example is the quarternary majorityfunction : { , }
7→ { , } (see details e.g. in [KDS]).Let us denote by H scp ( f ) the minimum number of subcubes in a subcube partitionwhich computes the Boolean function f .1.1. Prior work.
In the last decades there are several interplay between complexitytheory and additive combinatorics. One of the most interesting example is connectionbetween notions in computer sciences and the
Gowers norm (see e.g. [ST], [TR]).Another interesting example is an additive communication complexity problem whichis supported by an example of Behrend on the maximal density of a set not containingthree-term arithmetic progression (see e.g. [RY]).2.
Result
The aim of this note is to prove the following uncertainty estimation related to degf , the degree of f , H scp ( f ) and the additive energy E ( A ): Theorem 2.1.
Let f be any Boolean function, f : { , } n
7→ { , } with degree degf ,the set A its support i.e. A := f − (1) , H = H scp ( f ) , and E ( A ) its additive energy.We have the following uncertainity bound n n ≤ (8 degf deg f ) H · E ( A ) . NORBERT HEGYV ´ARI Preliminaries
Let f, g be two Boolean functions. The expected value of f is E ( f ) := 12 n X x ∈{ , } n f ( x ) , and the inner product of f and g is h f, g i := E ( f g ). For S ⊆ [ n ] the correspondinginput is x = ( x , x , . . . , x n ) ∈ { , } n namely x i = 1 if i ∈ S and x i = 0 otherwise.A basis function or character is defined by χ x ( y ) := ( − h x,y i , where h x, y i := P ni =1 x i y i (mod 2).For a set S ⊆ [ n ] the Fourier transform of f is b f ( S ) = h f, χ S i .For the Fourier transform the following are true:( i ) h f, g i = X r ∈{ , } n b f ( r ) b g ( r ) (Plancherel)( ii ) k f k = E ( f ) = h f, f i = X r ∈{ , } n b f ( r ) (Parseval)So by the Parseval formula for the indicator function we have P r ∈{ , } n b A ( r ) = n | A | .For functions f and g their convolution is defined by f ∗ g ( x ) := E f ( y ) g ( x + y ) . It is easy to verify that the convolution is associative: f ∗ ( g ∗ h ) = ( f ∗ g ) ∗ h .We will use the notation | X | ≪ | Y | to denote the estimate | X | ≤ C | Y | for someabsolute constant C > N UNCERTAINTY INEQUALITIES RELATED TO SUBCUBE PARTITIONS AND ADDITIVE ENERGY5 Proof
For the proof we need some lemmas.
Lemma 4.1. (1) E x,y,z ( f ( x ) f ( y ) f ( z ) f ( x + y + z )) = X r b f ( r ) . This statement can be found for example in [D, p.22] without proof; so for thesake of completeness we include a short proof.
Proof.
Using that E z ( f ( z ) f ( x + y + z )) = f ∗ f ( x + y ), we have E x,y,z ( f ( x ) f ( y ) f ( z ) f ( x + y + z )) = E x ( f ( x ) E y ( f ( y ) E z ( f ( z ) f ( x + y + z )))) == E x ( f ( x ) E y ( f ( y ) f ∗ f ( x + y ))) = E x ( f ( x )( f ∗ ( f ∗ f ( x )))) . Write briefly f ∗ ( r ) instead of ( f ∗ ( f ∗ f ))( r ). By the Plancherel formula, theassociative of the convolution, and the Fourier transformation of a convolution wehave E x,y,z ( f ( x ) f ( y ) f ( z ) f ( x + y + z )) = X r [ \ f · f ∗ ( r )] == X r b f ( r ) \ f ∗ ( r ) = X r b f ( r ) b f ( r ) = X r b f ( r ) . (cid:3) Corollary 4.2.
Let A := f − (1) . Then P r b f ( r ) = n E ( A ) .Proof. As we detected P r b f ( r ) can be written as E x,y,z ( f ( x ) f ( y ) f ( z ) f ( x + y + z )).Since in F x + ( x + y + z ) = y + z holds, thus we have E x,y,z ( f ( x ) f ( y ) f ( z ) f ( x + y + z )) == 12 n |{ ( a , a , a , a ) ∈ A : a + a = a + a }| = 12 n E ( A ) . (cid:3) NORBERT HEGYV ´ARI
The key step of the proof is to give a lower and an upper bound for the totalinfluence.First recall that I ( f ) = X S ∈ [ n ] | S | b f ( S ) . which can easily be proven. Lemma 4.3. I ( f ) ≤ n ( degf ) / k f k / ( E ( A )) / . Proof.
By the H¨older inequality I ( f ) = X S ∈ [ n ] | S | b f ( S ) = X S ∈ [ n ] ( | S | b f ( S )) / | b f ( S ) | / ≤≤ X S ∈ [ n ] | S || b f ( S ) | / X S ∈ [ n ] | b f ( S ) | / ≤ Now using Corollary 4.2 and ( P S ∈ [ n ] | S || b f ( S )) / ≤ ( degf ) / ( P S ∈ [ n ] | b f ( S ) | ) / , wehave ≤ n ( degf ) / k f k / ( E ( A )) / as we stated. (cid:3) The lower bound for the total influence comes from the folklore; since
Inf i ( f ) = P r x ∈{ , } n [ f ( x ) = f ( x + e i )], using Schwartz-Zippel lemma one can show, that Inf i ( f ) ≥ degf and hence(2) I ( f ) ≥ n degf (Maybe the first explicit estimation can be found in [NSZ]).In the rest of the paper we recall some behaviour of the subcube partition tocomplete the proof of the theorem. N UNCERTAINTY INEQUALITIES RELATED TO SUBCUBE PARTITIONS AND ADDITIVE ENERGY7
Now let C , C , . . . , C H be a minimal subcube partition of { , } n which computes f . Let us denote by f i the value of f in the part C i . So if i is the indicator functionof C i then clearly f ( x ) = P Hi =1 f i i ( x ) and hence by the linearity we have b f ( S ) = H X i =1 f i [ i ( S ) . It is well-known that if for a function g ( x ) ≤ x ∈ { , } n holds then b g ( S ) ≤ U ⊆ { , } n | \ U ( S ) | = (cid:12)(cid:12)(cid:12) n X T U ( T )( − ) | S ∩ T | (cid:12)(cid:12)(cid:12) ≤ n X T | U ( T ∩ U )( − ) | S ∩ T | | ≤ (3) ≤ n X T ⊆ U n −| U | = 1 . Thus using this bound for sets U = C i ; i = , , . . . H we have k f k = X S (cid:12)(cid:12) b f ( S ) (cid:12)(cid:12) ≤ H X i =1 | f i [ i ( S ) | ≤ H. Finally by (2), Lemma 4.3 and the calculation above we get n degf ≤ I ( f ) ≤ n ( degf ) / k f k / ( E ( A )) / ≤ n ( degf ) / H / ( E ( A )) / . Comparing the LHS and RHS and rearranging the inequality we obtain the desiredestimation.
NORBERT HEGYV ´ARI Concluding remarks
1. Let us first remark that there is a refinement of the theorem if we have aninformation on the cardinality of the subcubes. For instance when the cardinalitiesare concentrated to the ”middle size”: assume, there are parameters η, ν ∈ (0 , η ≤ | C i | /n ≤ ν holds for every i = 1 , , . . . H . Then the bound for theFourier transform of the indicators instead of (3) will be | [ i ( S ) | ≤ n | C i | · X k ≤ n − ηn (cid:18) nk (cid:19) . Now using the bound for sets C i and using the well-known estimates for binomialcoefficients (where h ( x ) is the binary entropy function defined by h ( x ) = − x log x − (1 − x ) log(1 − x )) εn X k =0 (cid:18) nk (cid:19) ≤ nh ( ε ) ε ∈ [0 , εn ∈ N we obtain a stronger bound in the theorem. Namely the factor H / should changeto (2 ( ν + h (1 − η ) − n H ) / which is less than the original one for middle concentrated parts.2. The calculated bound at the end of the proof of theorem we achieved I ( f ) ≤ n ( degf ) / H / ( E ( A )) / . Now let us introduce the entropy type quantity E ( g ) := E ( g ) log(1 / E ( g )). A classicalresult of Harper, Bernstein, Lindsey and Hart says I ( g ) ≥ E ( g ) (see e.g. [KF]).In E ( g ), g = µ ( A ) is the density of the set A , i.e. one can read this entropy as E ( µ ( A )) := E ( µ ( A )) log(1 / E ( µ ( A ))).So one can conclude the following uncertainity inequality too: N UNCERTAINTY INEQUALITIES RELATED TO SUBCUBE PARTITIONS AND ADDITIVE ENERGY9
Proposition 5.1. n +3 E ( µ ( A )) ≤ ( degf ) H E ( A ) . Acknowledgement.
This work is supported by NKFIH (OTKA) grant K-129335.
References [D] Ryan O’Donnell, Analysis of Boolean Functions, Cambridge University Press, 2014[KF] N.Keller, N. Lifshitz, Approximation of biased Boolean functions of small total influence byDNFs, Bulletin of the London Mathematical Society, (2018), p. Vol. 50, (4) p.667-679[KDS] R. Kothari, D. Racicot-Desloges, M. Santha, Separating decision tree complexity from sub-cube partition complexity , In Proceedings of 19th International Workshop on Randomizationand Computation, 2015.[NSZ] Nisan, N., Szegedy, M. On the degree of boolean functions as real polynomials. ComputComplexity 4, 301313 (1994). https://doi.org/10.1007/BF01263419[RY] A. Rao, A. Yehudayoff, Communication Complexity,https://homes.cs.washington.edu/ anuprao/pubs/book.pdf[ST] A. Samorodnitsky, L. Trevisan, Gowers Uniformity, Influence of Variables, and PCPs,STOC’06., arXiv:math/0510264v1 [math.CO][TV] T.Tao, V.Vu: Additive Combinatorics, Cambridge University Press, Cambridge 2006[TR] L. Trevisan, Earliest Connections of Additive Combinatorics and Computer Science,available in https://lucatrevisan.wordpress.com/2009/04/17/earliest-connections-of-additive-combinatorics-and-computer-science/
Norbert Hegyv´ari, ELTE TTK, E¨otv¨os University, Institute of Mathematics, H-1117 P´azm´any st. 1/c, Budapest, Hungary and Alfr´ed R´enyi Institute of Mathe-matics, Hungarian Academy of Science, H-1364 Budapest, P.O.Box 127.
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