A Note on the Entropy/Influence Conjecture

A Note on the En trop y/Influence Conj ecture Nathan Keller ∗ , Elch a nan Mossel † , and T omer Schlank ‡ Octob er 22, 2018 Abstract The ent r opy/influence conjecture, ra ised by F riedgut and K alai [8] in 1 996, seek s to rela te t wo different measur es of concentration of the F o ur ier co efficients of a Bo o lean function. Roughly saying, it cla ims that if the F ourier sp ectr um is “ smeared out”, then the F ourier co efficients are concentrated on “hig h” levels. In this note w e g eneralize the conjecture to biased pr o duct measure s o n the disc r ete cub e, a nd prove a v aria nt of the conjecture for functions with an extr emely low F ourier weight on the “hig h” levels. 1 In tro d u ction Definition 1.1. Consider the discr ete cub e { 0 , 1 } n endowe d with the pr o duct me asur e µ p = ( pδ { 1 } + (1 − p ) δ { 0 } ) ⊗ n , denote d in the se quel by { 0 , 1 } n p , and let f : { 0 , 1 } n p → R . The F ourier- Walsh exp ansion of f with r esp e ct to the me asur e µ p is the unique exp ansion f = X S ⊂{ 1 , 2 ,. ..,n } α S u S , wher e for any T ⊂ { 1 , 2 , . . . , n } , 1 u S ( T ) =  − r 1 − p p  | S ∩ T |  r p 1 − p  | S \ T | . In p articular, for the uniform me asur e (i.e., p = 1 / 2 ), u S ( T ) = ( − 1) | S ∩ T | . The c o efficients α S ar e denote d by ˆ f ( S ) , 2 and the level of the c o efficie nt ˆ f ( S ) is | S | . Prop erties of the F ourier-W alsh expansion are one of the main ob jects of s tu dy in d iscr ete harmonic analysis. The entrop y/influence conjecture, raised b y F riedgut and Kalai [8] in 1996, seeks to relate t w o measures of concen tration of the F ourier co efficien ts (i.e. coefficients of the F ourier-W alsh expansion) of Bo olean functions. The first of them is the sp e ctr al entr opy . ∗ W eizmann Institute of Science. P artially supp orted by the Koshland Center for Basic R esarc h. E-mail: nathan.keller@w eizmann.ac.il. † U.C. Berk eley and W eizmann In stitute of Science. Supp orted by DMS 0548249 (CAREER) aw ard, by DOD ONR grant N0001411 10140, b y ISF gran t 1300/08 and by a Minerv a Grant. E-mail: mossel@stat.b erkel ey .edu. ‡ Hebrew Universit y . P artially su p p orted by the Hoffman program for leadership. E-mail: tomer.sc hlank@ gmail.com. 1 Throughout th e pap er, we identi fy elements of { 0 , 1 } n with subsets of { 1 , 2 , . . . , n } in the natural wa y . 2 Note that since the functions { u S } S ⊂{ 1 ,...,n } form an orthonormal basis, th e representati on is ind eed un ique, and the coefficients are given by the formula ˆ f ( S ) = E µ p [ f · u S ] . 1 Definition 1.2. L et f : { 0 , 1 } n p → {− 1 , 1 } b e a Bo ole an function. The sp e ctr al entr opy of f with r esp e ct to the me asur e µ p is En t p ( f ) = X S ⊂{ 1 ,. ..,n } ˆ f ( S ) 2 log 1 ˆ f ( S ) 2 ! , wher e the F ourier-Walsh c o efficients ar e c ompute d w.r.t. to µ p . Note th at by P arsev al’s identit y , for any Bo olean fun ction we h a ve P S ˆ f ( S ) 2 = 1, and thus, the squares of the F ourier co efficien ts can b e view ed as a p robabilit y d istribution on the set { 0 , 1 } n . In this notation, the sp ectral en tr op y is simply th e en tr opy of this distribution, and in tuitivel y , it measures ho w m uch are the F ourier coefficients “smeared out”. The second notion is the total influenc e . Definition 1.3. L et f : { 0 , 1 } n p → { 0 , 1 } . F or 1 ≤ i ≤ n , the influenc e of the i -th c o or dinate on f with r esp e ct to µ p is I p i ( f ) = Pr x ∼ µ p [ f ( x ) 6 = f ( x ⊕ e i )] , wher e x ⊕ e i denotes the p oint obtaine d fr om x by r eplacing x i with 1 − x i and le aving the other c o or dinates unchange d. The total influenc e of the function f is I p ( f ) = n X i =1 I p i ( f ) . Influences of v ariables on Bo olean functions we r e studied extensiv ely in the last decades, and h a ve applications in a wide v ariet y of fields, in cluding T heoretical C omputer S cience, Com bin atorics, Mathematical Ph ysics, So cial Ch oice Theory , etc. (see, e.g., the sur vey [10].) As observed in [9 ], the tota l infl uence can b e expressed in terms of the F ourier coefficients: Observ ation 1.4. L et f : { 0 , 1 } n p → {− 1 , 1 } . Then I p ( f ) = 1 4 p (1 − p ) X S | S | ˆ f ( S ) 2 . (1) In p articular, for the uniform me asur e µ 1 / 2 , I 1 / 2 ( f ) = P S | S | ˆ f ( S ) 2 . Th us, in terms of the d istr ibution induced by the F ourier coefficien ts, the total influence is (up to normalization) the exp e ctation of the level of the coefficients, and it measures the question w hether the coefficien ts are concen tr ated on “high” lev els. The entrop y/influence conjecture asserts the follo wing: Conjecture 1.5 (F riedgut and Kalai) . Consider the discr ete cub e { 0 , 1 } n endowe d with the uniform me asur e µ 1 / 2 . Ther e exists a universal c onstant c , such that for any n and for any Bo ole an f unction f : { 0 , 1 } n 1 / 2 → {− 1 , 1 } , En t 1 / 2 ( f ) ≤ c · I 1 / 2 ( f ) . 2 The conjecture, if confirmed, has numerous significant implications. F or example, it wo u ld imply that for an y prop erty of graphs on n ve r tices, the sum of influ ences is at least c (log n ) 2 (whic h is tight for the prop ert y of con taining a clique of size ≈ log n ). The b est currently kno w n lo we r b ound , by Bourgain and Kalai [5], is Ω((lo g n ) 2 − ǫ ), for an y ǫ > 0. Another consequence of the co n jecture w ould b e an affirmativ e answe r to a v arian t of a conjecture of Mansour [13] stat in g that if a Bo olean function can b e repr esen ted b y a DNF form u la of p olynomial size in n (the num b er of co ordinates), then most of its F ourier weigh t is concen trated on a p olynomial num b er of coefficien ts (see [14] for a detailed explanation of th is application). This conjecture, raised in 1995 , is still wide op en. In this note we explore the en tropy/ in fl uence conject ure in tw o dir ections: Biased measure on the discrete cub e. W e state a generalization of the conjecture to the pro du ct measur e µ p on the discrete cub e: Conjecture 1.6. Ther e exists a universal c onstant c , su c h that for any 0 < p < 1 , for any n and for any Bo ole an function f : { 0 , 1 } n p → {− 1 , 1 } , En t p ( f ) ≤ cp log(1 /p ) · I p ( f ) . W e pro ve that Conjecture 1.6 follo ws from the original Entrop y/Influ ence conjecture, and that it is tight for the graph prop ert y of cont aining a clique of fixed s ize (at th e critica l p roba- bilit y). Th is answers a qu estion raised b y K alai [11]. F unctions w it h a low F ourier weigh t on t he “high” lev els. W e consider a w eake r v ersion of the conjecture s tating th at if “almost all” the F our ier w eight of a function is concen trated on the lo west k level s , then its en tropy is at most c · k . W e p ro ve this state m en t in an extreme case: Prop osition 1.7. L et f : { 0 , 1 } n 1 / 2 → {− 1 , 1 } b e a Bo ole an function such that al l the F ourier weight of f is c onc entr ate d on the first k leve ls. Then al l the F ourier c o effici ents of f ar e of the form ˆ f ( S ) = a ( S ) · 2 − k wher e a ( S ) ∈ Z . In p articular, En t 1 / 2 ( f ) ≤ 2 k . Then w e cite a stronger unpu blished result of Bourgain and Kalai [6] wh ic h sho w s that if the F ourier weigh t b ey ond the k th lev el deca y s exp onen tially , then the sp ectral en tropy is b ounded from ab o ve by c · k . Finally , we suggest that if one could generalize the result of Bourgain and Kalai to some slo we r rate of deca y , th is w ould lead to a pro of of the ent ir e en trop y/influ ence conjecture, using a tensorisation tec h nique. This n ote is orga nized as follo ws. In S ection 2 w e consider the generali zation of the en- trop y/influ ence conjecture to the biased measure on th e discrete cub e. F u nctions with a low F ourier weig ht on the h igh lev els are discussed in Section 3. W e conclude the pap er with an easy pro of of a w eak er upp er b ound on the entrop y , and with a co n nection betw een the en tropy/influence conjecture and F r iedgut’s c h aracterizat ion of functions with a lo w total in- fluence [7] in Section 4. 3 2 En trop y/Influence Conjecture for the Pro duct Measure µ p on the Discrete Cub e In th is s ection w e consider the space { 0 , 1 } n p , for 0 < p < 1. First we f orm ulate a v arian t of th e en tropy/influence conjecture f or th e b iased measur e and pr o ve that it follo ws from the original conjecture. T hen w e sho w that it is tigh t for the graph pr op ert y of conta in ing a cop y of a complete graph K r as an ind uced su bgraph, f or random graph s distrib uted according to the mo del G ( n, p ), at the critic al p robabilit y p c . Prop osition 2.1. Assume that the entr opy/influenc e c onje ctur e holds . Then ther e exists a universal c onstant c such that for any 0 < p < 1 , for any n and for any f : { 0 , 1 } n p → {− 1 , 1 } , we have En t p ( f ) ≤ cp log(1 /p ) · I p ( f ) . Our pro of is based on a standard reduction fr om the b iased measure µ p to the u niform measure µ 1 / 2 first considered in [4 ]. Let p ≤ 1 / 2, and assu m e that p = t/ 2 m . 3 F or any function f : { 0 , 1 } n → R w e define a function Red ( f ) = g : { 0 , 1 } mn → R a s f ollo ws: eac h y ∈ { 0 , 1 } mn is considered as a concatenation of n v ectors y i ∈ { 0 , 1 } m , and eac h suc h ve ctor is translated to a n atural num b er 0 ≤ B in ( y i ) < 2 m through its b inary exp ansion (i.e., B in ( y i ) = P m − 1 j =0 2 j · y i m − j ). Then, for an y y ∈ { 0 , 1 } mn , g ( y ) = g ( y 1 , y 2 , . . . , y n ) := f  h ( y 1 ) , h ( y 2 ) , . . . , h ( y n )  , where h : { 0 , 1 } m → { 0 , 1 } is give n by h ( y i ) =  1 , B in ( y i ) ≥ 2 m − t 0 , B in ( y i ) < 2 m − t. W e u se tw o simp le prop erties of the redu ction. T he first, pro v ed b y F riedgut and Kalai [8], relates th e total in fluence of g (w.r.t. µ 1 / 2 ) to th at of f (w .r .t. to µ p ). Lemma 2.2 (F riedgut and Kalai) . L et f : { 0 , 1 } n p → {− 1 , 1 } , and let g = Red ( f ) . Then I 1 / 2 ( g ) ≤ 6 p ⌊ log(1 /p ) ⌋ I p ( f ) . (2) The second prop ert y relates the F ourier co efficien ts of f (w.r.t. µ p ) to corresp onding co ef- ficien ts of g (w.r.t. µ 1 / 2 ). Lemma 2.3. L et f : { 0 , 1 } n p → R , a nd let g = R ed ( f ) . F or any S ⊂ { 1 , 2 , . . . , mn } , deno te S i = S ∩ { ( i − 1) m + 1 , ( i − 1) m + 2 , . . . , im } , and for S ′ ⊂ { 1 , 2 , . . . , n } , let V ( S ′ ) = { S ⊂ { 1 , 2 , . . . , mn } : { i : | S i | > 0 } = S ′ } . Then: X S ∈ V ( S ′ ) ˆ g ( S ) 2 = ˆ f ( S ′ ) 2 . (3) 3 It is clear that there is no loss of generality in assuming that p is diadic, as the results for general p follow immediately by approximatio n . 4 Pro of: F or eac h S ′ ⊂ { 1 , 2 , . . . , n } , let f S ′ : { 0 , 1 } n p → R b e defined by f S ′ = ˆ f ( S ′ ) u S ′ . W e claim that Red ( f S ′ ) = X S ∈ V ( S ′ ) ˆ g ( S ) u S . (4) This claim implies the assertion, as b y the P arsev al iden tit y , Equation (4) imp lies: X S ∈ V ( S ′ ) ˆ g ( S ) 2 = || Red ( f S ′ ) || 2 2 = || f S ′ || 2 2 = ˆ f ( S ′ ) 2 . (The fir st and third equalities use th e Parsev al id entit y , and the middle equalit y h olds since by the str ucture of the reduction, it preserve s all L p norms.) In order to pro ve Equation (4), w e us e Prop osition 2.2 in [12 ] that d escrib es the exact relatio n b et ween the F ourier co efficien ts of Red ( f ) and the corresp ondin g coefficients of f . By the prop osition, for all S ∈ V ( S ′ ), \ Red ( f )( S ) = c ( S, p ) · ˆ f ( S ′ ) , where c ( S, p ) dep ends on S and p but not on f . Hence, for all S ∈ V ( S ′ ), we ha ve \ Red ( f S ′ )( S ) = \ Red ( f )( S ) (since b oth are determined b y S, p , and ˆ f ( S ′ )). Similarly , for all S 6∈ V ( S ′ ), \ Red ( f S ′ )( S ) = 0, s ince c f S ′ ( S ′′ ) = 0 for all S ′′ 6 = S ′ . Therefore, the F our ier expansion of Red ( f S ′ ) is: Red ( f S ′ ) = X S ∈ V ( S ′ ) \ Red ( f )( S ) u S , as asserted.  No w w e are ready to pro ve Prop osition 2.1. Pro of: Let f : { 0 , 1 } n p → {− 1 , 1 } , and let g = R ed ( f ). By Equ ation (3), En t 1 / 2 ( g ) = X S ⊂{ 1 , 2 ,. ..,mn } ˆ g ( S ) 2 log 1 ˆ g ( S ) 2 = X S ′ ⊂{ 1 , 2 ,...,n } X S ∈ V ( S ′ ) ˆ g ( S ) 2 log 1 ˆ g ( S ) 2 ≥ X S ′ ⊂{ 1 , 2 ,...,n } X S ∈ V ( S ′ ) ˆ g ( S ) 2 log 1 ˆ f ( S ′ ) 2 = X S ′ ⊂{ 1 , 2 ,...,n } ˆ f ( S ′ ) 2 log 1 ˆ f ( S ′ ) 2 = Ent p ( f ) . (5) Com bin ing Equation (5) with Equ ation (2) and applying the entrop y/influence conjecture to g , w e get: En t p ( f ) ≤ Ent 1 / 2 ( g ) ≤ c · I 1 / 2 ( g ) ≤ c · 6 p ⌊ log (1 /p ) ⌋ I p ( f ) , and therefore, En t p ( f ) ≤ c ′ p log (1 /p )I p ( f ) , as asserted.  Consider the r andom graph mo d el G ( n, p ). Recall that in this mo del, the probabilit y s pace is { 0 , 1 } N p , where N =  n 2  , the co ordinates corresp ond to the edges of a graph on n v ertices, and eac h ed ge exists in the graph with probabilit y p , indep en den tly of the other edges. It is w ell-kno wn that for the graph pr op ert y of con taining the complete graph K r as an induced subgraph, there exists a th r eshold at p t = Θ( n − 2 / ( r − 1) ). This means that if p << n − 2 / ( r − 1) then 5 Pr[ K r ⊂ G | G ∈ G ( n, p )] is close to zero, and if p >> n − 2 / ( r − 1) then Pr[ K r ⊂ G | G ∈ G ( n, p )] is close to one. W e choose a v alue p 0 in the critical r ange, consider the c haracteristic fu nction f of this graph p rop erty in G ( n, p 0 ), and sh ow that the assertion of Prop osition 2.1 is tight for f . In ord er to simplify the computation, we choose p 0 suc h that the exp ected n u m b er of copies of K r in G ( n, p 0 ) is “nice”. Ho wev er, the same argum en t holds for an y v alue of p in the critical range. Prop osition 2.4. L et n, r b e i nte gers such that r < log n . Consider the r andom gr aph G ( n, p 0 ) wher e p 0 is chosen such that  n r  · p ( r 2 ) 0 = 1 / 2 . L et f b e define d by: f ( G ) = 1 ⇔ G c ontains a c opy of K r as an induc e d sub gr aph , and f ( G ) = 0 otherwise. Then En t p 0 ( f ) ≥ c · p 0 log(1 /p 0 ) · I p 0 ( f ) , wher e c is a universal c onstant. Pro of: The result is a com bination of an upp er b ound on I p 0 ( f ) with a lo wer b ound on En t p 0 ( f ). In ord er to b ound I p 0 ( f ) from ab ov e, note th at a necessary (but not s u fficien t) condition for an edge e = ( v , w ) to b e pivot al f or f at a graph G 4 is that there exists a set S of r vertices including v and w such that all  r 2  edges inside S except for e app ear in G . Hence, a simple union b ound yields that for any edge e , I p 0 e ( f ) ≤  n − 2 r − 2  · p ( r 2 ) − 1 0 = r ( r − 1) n ( n − 1) p 0 ·  n r  p ( r 2 ) 0 = r ( r − 1) 2 n ( n − 1) p 0 , and thus, I p 0 ( f ) = X e I p 0 e ( f ) ≤ 1 p 0 · r ( r − 1) 4 . (6) In order to boun d Ent p 0 ( f ) from b elo w, w e sho w that a t least a constant p ortion of the F ourier weig h t of f is concen trated on co efficien ts that corresp ond to copies of K r in { 0 , 1 } N . Concretely , w e sho w that if S corresp ond s to a cop y of K r , then : ˆ f ( S ) 2 ≥ c ′ ·  n r  − 1 . (7) As the n umb er of suc h co efficients is  n r  , it will follo w that: En t p 0 ( f ) ≥ X { S : S is a copy of K r } ˆ f ( S ) 2 log 1 ˆ f ( S ) 2 ! ≥ c ′ · log  n r  ≥ c ′′ · r log( n ) , (8) where the righ tmost inequalit y holds sin ce r < log n . Finally , a com bination of Equation (8) with Equ ation (6) w ill imp ly: En t p 0 ( f ) ≥ c ′′ · r log( n ) ≥ c ′′ · r ( r − 1) 2 · log (1 /p 0 ) ≥ c ′′ · p 0 log(1 /p 0 ) · I p 0 ( f ) , 4 An edge e is pivotal for the prop erty f at a graph G if f ( G ) = 1 and f ( G \ { e } ) = 0. 6 as asserted. T o pro ve E quation (7), consider a sp ecific cop y H of K r and denote it s set of edges by S = E ( H ). By the definition of the F ourier coefficient s , w e ha v e: ˆ f ( S ) = X T ∈{ 0 , 1 } N µ p 0 ( T )  − r 1 − p 0 p 0  | S ∩ T |  r p 0 1 − p 0  ( r 2 ) −| S ∩ T | f ( T ) = X T ∈{ 0 , 1 } N µ p 0 ( T \ S ) · ( p 0 (1 − p 0 )) ( r 2 ) / 2 ( − 1) | S ∩ T | f ( T ) =( p 0 (1 − p 0 )) ( r 2 ) / 2 X T ∈{ 0 , 1 } N µ p 0 ( T \ S )( − 1) | S ∩ T | f ( T ) , (9) where µ p 0 ( T \ S ) denotes the ind uced measure of the graph T \ S . Note th at the total con trib ution to ˆ f ( S ) of { T ∈ { 0 , 1 } N : S ⊂ T } is ( − 1) | S | · ( p 0 (1 − p 0 )) ( r 2 ) / 2 , ( 10) since f ( T ) = 1 for all T ⊃ S . On the other hand , if f ( T ) = 1 and T ) S , then T con tains a cop y of K r , in which k ≤ r − 1 v ertices are included in V ( H ), and the r emaining r − k v ertices are not included in V ( H ). Hence, th e total con tribution to ˆ f ( S ) of { T ∈ { 0 , 1 } N : S ( T } is b ound ed fr om ab o ve (in absolute v alue) b y: ( p 0 (1 − p 0 )) ( r 2 ) / 2 · r − 1 X k =0  n r − k  r k  p ( r 2 ) − ( k 2 ) 0 = ( p 0 (1 − p 0 )) ( r 2 ) / 2 · (1 / 2 + o n (1)) , (11) since for our choic e of p 0 , the term corresp onding to k = 0 equals  n r  p ( r 2 ) 0 = 1 / 2, and the other terms are negligible. Com bining estimates (10) and (11), we get: ˆ f ( S ) 2 ≥ (1 − 1 / 2 − o n (1)) 2 ( p 0 (1 − p 0 )) ( r 2 ) ≥ cp ( r 2 ) 0 = ( c/ 2) ·  n r  − 1 . (12) This completes the pr o of.  W e conclude this section by n oting that if p is inv ers e p olynomially s m all as a fun ction of n , then one can easily prov e a statemen t whic h is only slightly w eaker than th e entrop y/influence conjecture. In [14] it was shown that w ith resp ect to the uniform measur e, w e ha ve En t 1 / 2 ( f ) ≤ (log n + 1)I 1 / 2 ( f ) + 1 , for any Bo olean f . The statemen t generalizes easily to a general biased measure µ p , and yields the f ollo w ing: Claim 2.5. Ther e exists a universal c onstant c such that f or any 0 < p < 1 , for any n and for any f : { 0 , 1 } n p → {− 1 , 1 } , we have En t p ( f ) ≤ cp (1 − p ) log ( n ) · I p ( f ) . 7 Pro of: Assume w.l.o.g . th at p ≤ 1 / 2. As sh o wn in [14], we ha ve: En t p ( f ) ≤ (log n + 1) X S | S | ˆ f ( S ) 2 + ǫ log (1 /ǫ ) + 2 ǫ, where 1 − ǫ = ˆ f ( ∅ ) 2 . (Not e that this part of the pr o of of Theorem 3.2 in [14] h olds without an y change for the b iased measure). In order to b ound the term ǫ log (1 /ǫ ) + 2 ǫ , it w as sho wn in Prop osition 3.6 of [14] that by the edge isop erimetric inequalit y on the cub e, it is b ounded from ab o ve by 2I 1 / 2 ( f ). By Equation (2), this implies that for the measure µ p , we h a ve ǫ log (1 /ǫ ) + 2 ǫ ≤ 12 p ⌊ log (1 /p ) ⌋ I p ( f ) (since the r eduction from the biased measure to the u niform measure p r eserv es the exp ectation). Th us, by Equation (1), En t p ( f ) ≤ (log n + 1) · 4 p (1 − p )I p ( f ) + 12 p ⌊ lo g (1 /p ) ⌋ I p ( f ) ≤ cp (1 − p ) log ( n )I p ( f ) , as asserted.  F or p that is inv erse p olynomially small in n , the statemen t of Claim 2.5 differs from the assertion of the ent r op y/infl u ence conjecture only by a constant factor. 3 F unctions with a Lo w F ourier W e igh t on the High Lev els In this section we consider the uniform measure µ 1 / 2 on th e discrete cub e, and stu dy Boolean functions with a lo w F ourier w eight on the high lev els. In ord er to simplify the expression of the F our ier expansion, w e r ep lace the d omain by {− 1 , 1 } n . As a resu lt, the charact er s are giv en b y the formula u { i 1 ,...,i r } ( x ) = x i 1 · x i 2 · . . . · x i r , and th us, the F ourier expansion of a function is simply its represen tation as a multiv ariate p olynomial. Prop osition 3.1. L et f : {− 1 , 1 } n 1 / 2 → Z , such that al l the F ourier weight of f is c onc entr ate d on the first k levels. Then al l the F ourier c o efficients of f ar e of the form ˆ f ( S ) = a ( S ) · 2 − k , wher e a ( S ) ∈ Z . In p articular, En t 1 / 2 ( f ) ≤ 2 k . Pro of: The pro of is b y induction on k . The case k = 0 is trivial. Ass u me that the assertion holds for all k ≤ d − 1, and let f b e a function of F ourier degree d (i.e., all its F our ier co efficien ts are concen tr ated on th e d lo we st lev els). F or 1 ≤ i ≤ n , let f i b e the discrete deriv ativ e of f with resp ect to the i th coordin ate, i.e., f i ( x 1 , . . . , x i − 1 , x i +1 , . . . , x n ) = f ( x 1 , . . . , x i − 1 , 1 , x i +1 , . . . , x n ) − f ( x 1 , . . . , x i − 1 , − 1 , x i +1 , . . . , x n ) 2 . It is easy to see that if f = P S ˆ f ( S ) u S , then the F ourier expansion of f i is given by: f i = X S ⊂ ( { 1 , 2 ,.. .,n }\{ i } ) ˆ f ( S ∪ { i } ) u S . (13) 8 Hence, f i is of F ourier degree at most d − 1. Note that b y the defin ition of f i , we ha ve 2 f i ( x ) ∈ Z for all x ∈ {− 1 , 1 } n − 1 , an d thus by the indu ction hypothesis, the F ou r ier co efficien ts of f i satisfy 2 ˆ f i ( S ) = a ( S ) · 2 − d +1 , where a ( S ) ∈ Z . This holds for an y 1 ≤ i ≤ n , and therefore, b y Equation (13), all the F ourier coefficients of f (except, p ossib ly , f or ˆ f ( ∅ )), are of the form ˆ f ( S ) = a ( S ) · 2 − d , where a ( S ) ∈ Z . Finally , ˆ f ( ∅ ) must b e also of this form, s in ce otherwise f ( x ) cannot b e an in teger. This complete s the pro of.  In an unpub lish ed work [6 ], Bourgain and Kalai obtained a stronger result: Theorem 3.2. L et f : {− 1 , 1 } n → {− 1 , 1 } , and assume that ther e exist c 0 > 0 , 0 < a < 1 / 2 , and k , such that for al l t , X { S : | S | >t } ˆ f ( S ) 2 ≤ e c 0 k · e − at , then for any α > 1 , ther e exists a set B α , such that: 1. log | B α | ≤ C · αk , wher e C dep ends only on a and c 0 . 2. P S 6∈ B α ˆ f ( S ) 2 ≤ n − α . The theorem asserts that if the F ourier w eight of f b eyo n d the k th lev el deca ys exp on en - tially , then most of the F ourier w eight of f is concen trated on exp ( C k ) co efficient s, and thus, En t 1 / 2 ( f ) ≤ C ′ k (for an appropriate c hoice of C ′ ). Th e pro of uses the d th d iscrete deriv ativ e of f (lik e our p ro of ab ov e), and the Bo n ami-Bec k n er h yp ercon tractiv e inequalit y [2, 1 ]. W e note that th e exact dep endence of C on a (i.e., the r ate of the exp onen tial deca y) in the as- sertion of the theorem, whic h is imp ortan t if a is allo wed to b e a f u nction of n , is of order C = Θ( a − 1 log( a − 1 )). A tensorisation techniq ue . In [11], Kalai observe d that th e en tropy/i n fluence conjecture tensorises, in the follo wing sense. F or f : {− 1 , 1 } l 1 / 2 → {− 1 , 1 } and g : {− 1 , 1 } m 1 / 2 → {− 1 , 1 } , define f ⊗ g : {− 1 , 1 } l + m 1 / 2 → {− 1 , 1 } b y: f ⊗ g ( x 1 , . . . , x l + m ) = f ( x 1 , . . . , x l ) · g ( x l +1 , . . . , x l + m ) . F urtherm ore, let f ⊗ N = f ⊗ f ⊗ . . . ⊗ f , where the tensorisation is p erformed N times. It is easy to see that I 1 / 2 ( f ⊗ N ) = N · I 1 / 2 ( f ) and En t 1 / 2 ( f ⊗ N ) = N · E nt 1 / 2 ( f ). Hence, p r o ving the entrop y/influence conjecture for an y “tensor p o wer” of f is equiv alen t to pro ving the conjecture for f itself. This observ ation w as used in [14 ] to d educe that it is su fficien t to pro ve a seemingly w eake r version of the conjecture: En t 1 / 2 ( f ) ≤ c I 1 / 2 ( f ) + o ( n ), wh er e n is the n umb er of v ariables. W e observe that tensorisation can b e used to enhance the r ate of deca y of the F ourier co efficien ts. By the La w of Large Numb ers, as N → ∞ , the lev el of the F our ier co efficien ts of f ⊗ N is concentrate d around its exp ectation, which is N · I 1 / 2 ( f ), and the rate of deca y ab o ve that leve l, i.e., P | S | >t d f ⊗ n ( S ) 2 b ecomes “almost” in verse exp onen tial in t . This holds ev en if th e r ate of d eca y of the F ourier co efficien ts of f is muc h slo wer (lik e the Ma jorit y function, for whic h P | S | >t \ M AJ ( S ) 2 ≈ t − 1 / 2 ). Therefore, if one obtains a result similar to Bourgain-Kalai’s Theorem 3.2 for a slo wer r ate of deca y , e.g., und er the weak er assumption P | S | >t ˆ f ( S ) 2 ≤ e c 0 √ k · e − a √ t , then the r esu lt can b e enhanced to any rate of deca y , b y tensoring the f unction to itself until its r ate of d eca y r eac hes e − a √ t . How ev er, w e weren’t able to fi nd suc h generalizat ion of the Bourgain-Kala i result. 9 4 Concluding Remarks W e conclude this pap er with t w o remarks relate d to the ent r op y/influ ence conjecture. A w eaker upp er b ound on the en trop y t hat can b e pro v ed ea sily . As mentio n ed in Section 2, it w as sho wn in [14] that with resp ect to the u niform m easure, one can easily prov e the f ollo w ing weak er upp er b ound on the en trop y of any Bo olean function: En t 1 / 2 ( f ) ≤ (log n + 1)I 1 / 2 ( f ) + 1 . W e pro vide an indep en d en t pr o of of a sligh tly stronger claim. Claim 4.1. F or any n and for any f : { 0 , 1 } n 1 / 2 → R , we have En t 1 / 2 ( f ) ≤ n X i =1 h (I 1 / 2 i ( f )) ≤ 2I 1 / 2 ( f )(log n − log I 1 / 2 ( f )) , wher e h ( x ) = − x log x − (1 − x ) log(1 − x ) . Pro of: As the pro of deals only with the un if orm measure on the discrete cub e, w e w rite En t( f ) and I( f ) instead of Ent 1 / 2 ( f ) an d I 1 / 2 ( f ) du r ing the p ro of. Let S ⊂ { 1 , 2 , . . . , n } b e c h osen according to the F ourier distribution (i.e., Pr[ S = S 0 ] = ˆ f ( S 0 ) 2 ), and let X i = 1 { i ∈ S } . Then by the basic r ules of entrop y , En t( f ) = H ( S ) = H ( X 1 , . . . , X n ) ≤ n X i =1 H ( X i ) = n X i =1 h (I i ( f )) , th us obtaining the fir st inequalit y . No te that if I i ( f ) ≥ 0 . 5, then h (I i ( f )) ≤ 2I i ( f ), and otherwise, h (I i ( f )) ≤ − 2I i ( f ) log I i ( f ). Therefore, 1 2 En t( f ) ≤ I( f ) + n X i =1 I i ( f )( − log I i ( f )) = I( f ) 1 + n X i =1 I i ( f ) I( f ) ·  − log I i ( f ) I( f ) − log I( f )  ! . W e n ote that the expression P n i =1 I i ( f ) / I( f )( − log I i ( f ) / I( f )) is the ent r op y of th e rand om v ariable Y defi ned by Pr[ Y = i ] = I i ( f ) / I( f ) whic h is supp orted on { 1 , 2 , . . . , n } , and is therefore b ound ed by log n . W e th u s conclude that 1 2 En t( f ) ≤ I( f )(1 + log n − log I( f )) as asserted.  It is easy to see that the b ound using the en tropy is stronger in some cases, in particular when there is v ariabilit y in the influences of differen t coord inates. W e note that the pro of d o es not use th e fact that f is Boolean and indeed it could n ot pro vid e a pro of of the Entrop y/Influence conjecture, as can b e seen, e.g., for the ma jorit y function, wh ere I 1 / 2 ( f ) is of order √ n wh ile P n i =1 h (I 1 / 2 i ( f )) is of order √ n log n . 10 Relation to F riedgut’s c haracterization of functions wit h a lo w influence sum. In [7], F riedgut sh o we d that any Bo olean fu n ction f : { 0 , 1 } n p → { 0 , 1 } essentiall y d ep ends on at most C ( p ) I( f ) co ordinates, where C ( p ) dep end s only on p . T he main step of the pro of is to sh o w that most of the F ourier weigh t of the fun ction is concent rated on sets that con tain one of these co ordinates. A stronger c laim one ma y hop e to prov e is that most of the F ourier w eight is concen trated on at most C ( p ) I( f ) co efficien ts. F orm ally , w e raise the follo w ing conjecture that resem bles the assertion of Bourgain-Kalai’s theorem: Conjecture 4.2. F or any 0 < p < 1 , ther e exists a c onstant C ( p ) > 0 such that for any ǫ > 0 , for any n and for any f : {− 1 , 1 } n p → {− 1 , 1 } , ther e exists a set B ǫ ⊂ { 0 , 1 } n such that: 1. log | B ǫ | ≤ C ( p ) · I( f ) , and 2. P S 6∈ B ǫ ˆ f ( S ) 2 < ǫ . This conjecture is clearly stronger than F riedgut’s theo r em and ev en imp lies a v ariant of Mansour’s conjecture [13] (since as s ho wn in [3], if a Bo olean fu nction f can b e r epresen ted by an m -term DNF, then I 1 / 2 ( f ) = O (log m ) ), but it still do es not imply the en trop y/influ ence conjecture, since the remaining F ourier co efficients (whose total F ourier wei gh t is at most ǫ ) can still con tribute n · ǫ to Ent 1 / 2 ( f ). 5 Ac kno wledgemen ts W e are grateful to Gil Kalai for introdu cing us to his unp ublished w ork with Jean Bourgain [6], and to Ry an O’Donnell for sending us h is recen t work [14]. References [1] W. Bec kn er, Inequalities in F ourier Analysis, Annals of Math. 102 (1975 ), pp. 159–182. [2] A. Bonami, Etud e des Co efficien ts F ourier des F onctiones de L p ( G ), Ann. Inst. F ourier 20 (1970 ), pp. 335–402. [3] R. Boppana, T he Av erage Sensitivit y of Bounded-Depth Circuits, Inf. Pr o c. L et. , 63(5) (1997 ), pp. 257–261. [4] J. Bourgain, J. Kahn, G. Kalai, Y. Katznelson, and N. Linial, The Influence of V ariables in Pro duct Spaces, Isr ael J. Math. 77 (1992), pp. 55–64. [5] J. Bourgain and G. Kalai, Infl uences of V ariables and Th reshold Inte rv als Under Group Symmetries, GAF A 7 (1997), pp. 438-461. [6] J. Bourgain and G. Kalai, unpu blished man u script, 2000. [7] E. F riedgut, Bo olean F unctions w ith Low Averag e Sensitivit y Dep end on F ew Co ordinates, Combinatoric a , 18(1) (1998), pp. 27–35. [8] E. F r iedgut and G. Kalai, Eve ry Monotone Graph Prop erty Has a Sharp Thresh old, P r o c. Amer. Math. So c. 124 (1996), p p. 2993–300 2. [9] J. Kahn, G. Kalai, and N. Linial, T he Infl uence of V ariables on Boolean F unctions, Pro c. 29-th Ann. Symp. on F oundations of Comp. Sci., pp . 68–80, Computer So ciet y Press, 1988. 11 [10] G. Kalai and M. Safra, Threshold Phenomena and In fluence, in: Computational Complexity and Statistic al Physics , (A.G. P ercu s , G. Istrate and C. Mo ore, ed s .), Oxford Universit y Press, New Y ork, 200 6, pp. 25-6 0. [11] G. Kalai, Th e Entrop y/Influ ence Conjecture, Blog en try , 2007. Av ailable on-line at: h ttp://terrytao.w ordp ress.com/2007 /08/16/gil-k a lai-the-ent r op yinfl uence-conjecture/ [12] N. K eller, A Simple Reduction fr om a Biased Measure on th e Discrete Cub e to the Uniform Measure, 2010. Av ailable on-line at: ht tp://arxiv.org/a b s/1001. 1167. [13] Y. Mansour , An O ( n log log n ) Learning Algorithm for DNF Under the Uniform Distribution, J. Comput. System Sci. 50 (1995), n o. 3, p p. 543–550. [14] R. O’Donnell, J. W right, and Y. Z hou, The F ourier Entrop y-Influ ence C on j ecture for certain classes of Bo olean functions, to app ear in ICALP’2011. Av ailable online at: h ttp://www.cs.cmu.edu/ o d onnell/pap ers/fei.p d f . 12