1. Introduction. The aim of this paper is to revisit some of the questions put forward in the paper [E-S] of Erdos and Szekeres.

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1 ON A PAPER OF ERDÖS AND SZEKERES J. BOURGAIN AND M.-C. CHANG ABSTRACT. Propositions..3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products n ( zaj, a a n integers. In the discussion below, {a,..., a n } will be either a proportional subset of {,..., n} or a set of large arithmetic diameter.. Introduction The aim of this paper is to revisit some of the questions put forward in the paper [E-S] of Erdos and Szekeres. Following [E-S], define M(a,..., a n = max z = n z a i (. where we assume a a 2 a n positive integers (in this paper, we restrict ourselves to distinct integers a < < a n. Denote f(n = It was proven in [E-S] that i= min M(a,...,a n and f (n = min M(a,..., a n. (.2 a a n a < <a n This lower bound remains presently still unimproved. In the other direction, [E-S] establish an upper bound Subsequent improvements were given by Atkinson [A] f(n 2n. (.3 f(n < exp(n c for some c > 0. (.4 f(n = exp{o(n 2 log n} (.5

2 2 J. BOURGAIN AND M.-C. CHANG and Odlyzko [O] f(n = exp{o(n 3 (log n 4/3 }. (.6 Also to be mentioned is a construction due to Kolountzakis ([Kol2], [Kol4] of a sequence < a < < a n < 2n + O( n for which f (n M(a,...,a n < exp{o(n 2 log n} (.7 (Note that Odlyzko s construction does not come with distinct frequencies. As shown by Atkinson [A], there is a relation between the [E-S] problem and the cosineminimum problem. Define M 2 (n = inf{ min θ with infinum taken over integer sets a < < a n. cosa j θ} (.8 Then log f (n < O(M 2 (n log n. (.9 The problem of determining M 2 (n was put forward by Ankeny and Chowla [C] motivated by questions on zeta functions. It is known that M 2 (n = O(n2 and conjectured by Chowla that in fact M 2 (n n 2 [C2]. The current best lower bound is due to Ruzsa [R] M 2 (n > exp(c log n (.0 for some c > 0. As pointed out in [O], polynomials of the form (. are also of interest in connection to Schinzel s problem [S] of bounding the number of irreducible factors of a polynomial on the unit circle in terms of its degree and L 2 -norm. Propositions. and.2 in this paper establish new results for dense sets S = {a < < a n }. The former improves upon (.7. Proposition.. There is a subset {a < < a n } {,...,N}, n N, such that 2 M(a,..., a n < exp(c n log n log log n. (.

3 ON A PAPER OF ERDÖS AND SZEKERES 3 On the other hand, the following holds Proposition.2. There is a constant τ > 0 such that if {a <... < a n } {,...,N} and n > ( τn, then M(a,...,a n > exp τn. (.2 The latter result generalizes the comment made in [E-S] that exists and is between and 2. lim [M(, 2,..., n n]/n (.3 In converse direction, one may prove new lower bounds on M(a,...,a n assuming that the set {a < < a n } has a sufficiently large arithmetic diameter. First, we are recalling the notion of a dissociated set of integers. We say that D = {ν,...,ν m } Z is dissociated provided D does not admit non-trivial 0,, relations. Thus implies ε ν + + ε m ν m = 0 with ε = 0,, (.4 ε = = ε m = 0. A more detailed discussion of this notion and its relation to lacunarity appears in 5 of the paper. Proposition.3. Assume {a < < a n } contains a dissociated set of size m. Then log M(a,...,a n m 2 ε (log n /2. (.5 Hence (.5 improves upon (.3 as soon as m (log n 3+ε. (.6

4 4 J. BOURGAIN AND M.-C. CHANG 2. Preliminary estimates Let z = e(θ = e 2πiθ. By taking the real part of Log( e 2πiθ = k= k e2πikθ, we have cos 2πkθ log z =. k Therefore, we have k= Fact. n z a j = e P n P k= cos 2πka j θ k. We first establish some preliminary inequalities for later use. Since the function e x is convex, we obtain for any probability measure µ on T that n e(a j θ µ e (P n P cos 2πka j k= µ(θ k and therefore we have Fact 2. n e(a j θ e min {P n P k= θ cos 2πka j k µ}(θ. Lemma 2.. log e 2πiθ J where ρ = J and (2. is valid for all θ. j cos 2πjθ + O ( J (2. Proof. We rely on a calculation that appears in [O], Proposition. Use the inequality ( [O], (2.4 eiθ 2 for θ [0, 2π], 0 < ρ <. (2.2 ρe iθ + ρ

5 ON A PAPER OF ERDÖS AND SZEKERES 5 From (2.2 by partial summation and since log e iθ log ρe iθ + log = log J 2 + ρ j cosjθ + log 2 + ρ j cos jθ + ρ 2 ( + ρ = log Thus (2. follows from (2.3 with ρ as above. J + C( ρ J( ρ ρ 2. (2.3 Proposition 2.2. There is a subset {a... a m } {,..., n} of size m n 2 and m z a k e c n log n(log log n. (2.4 L ( z = k= Remark. (2.4 is a slight improvement of the estimate m z a k e c nlog n k= L ( z = resulting from a construction in [Kol], p. 62 of a set {a,...,a m } as above and such that m cos 2πa k θ c m and Lemma 2. log k= m ( 2a k k= J j m k= C(logJ ( m J m + O < C log n n, ( m J cos 2πa k (jθ + O

6 6 J. BOURGAIN AND M.-C. CHANG taking J = m 2. Proof of Proposition 2.2. Take independent selectors (ξ j j<n with values 0, and mean E[ξ j ] = j. Let F n n(θ = 2 0<j<n ( j cos 2πjθ + be the Fejer kernel n m cosa k θ = ξ l cos lθ = 2 F n(θ 2 + (ξ l E[ξ l ] coslθ. (2.5 k= l= By Lemma 2. (applies with J = n 0 m log e 2πiakθ k= J m k= l= ( m J j cos 2πja kθ + O (2.6 and we take J at least n to bound the last term in the right hand side of (2.5 by n. We analyze the first term. Inserting (2.5 gives the sum of the following two expressions ((2.7 and (2.8 Since F n (jθ 0, (2.7 log J. Rewrite J (2.8 = J l= ( j 2 F n(jθ 2 (2.7 j (ξ l E[ξ l ] cos2πljθ. (2.8 [ J (ξ l E[ξ l ] l= Note that all frequencies in (2.9 are bounded by nj. j cos 2πjlθ ]. (2.9 Applying the probabilistic Salem-Zygmund inequality [Kol3] shows that with large probability (2.9 [ log nj l= Our next task is to evaluate the expression n A first observation is that we can assume J l= θ > 0n 2] j cos 2πjlθ 2. (2.0 J j cos 2πjlθ 2. (2.

7 Thus for l S R lθ < ρe(lθ < e cr =: ε. since otherwise ON A PAPER OF ERDÖS AND SZEKERES 7 e 2πia kθ 2πa k θ < 2π 0 < for all k =,..., m, and also the left hand side of (2.4 is bounded by. Next, we note that (since ρ = J Hence J l= j cos 2πjlθ log ρe(lθ ρ J + J( ρ J Fix θ and for < R log J define the dyadic set < log ρe(lθ +. j cos 2πjlθ 2 log ρe(lθ 2 + n. (2.2 l= S R = { l n : log ρe(lθ R}. Let q N be the smallest integer with qθ < 2ε. It follows that S R n q q > R 3, one obtains log ρe(lθ 2 ( n R 3 + R 2 +. Assuming l S R with collected contribution (summing over dyadic R n + (log J 2. (2.3 It remains to consider θ s with the property that for some large R and q < R 3, qθ < e cr. Hence either θ admits a rational approximation θ a < e cr < e cr, q < R 3 and (a, q = (2.4 q q or ( in (2.4 when a = 0, by (2. n θ < e cr. (2.5

8 8 J. BOURGAIN AND M.-C. CHANG Consider first the case (2.5. Then S R {l =,...,n : lθ < e cr } ne cr and the above estimate still holds. Assume next that θ satisfies (2.4. Write θ = a q + ψ with β = ψ < e cr. (2.6 First, we consider the case β nq. Let V {,..., n} be an interval of size so that {lθ : l V } consists of qβ-separated qβ points filling a fraction of [0, ] (mod. Hence log ρe(lθ 2 βq l V 0 βq + log2 J log ρe(t 2 dt + log 2 ( ρ and unless log ρe(lθ 2 n + nq β log 2 n n l= qβ log 2 n >, i.e. log n > e cr or R log log n where we used (2.4. Thus if β nq, (2.2 n(log log n2. The next case is β < 00nq. It follows that for l n We obtain lθ la q log ρe(lθ 2 n q l < 00q. (2.7 0 log ρe(t 2 dt n

9 ON A PAPER OF ERDÖS AND SZEKERES 9 and log ρe(lθ 2 qβ q l nβ 0 log ρe(t 2 dt nβ ( log 2dt qβ 0 t n q (log nβ2. (2.8 We obtain again a bound O(n unless lognβ > q i.e. q β < e n. (2.9 Thus (2.7 may be replaced by lθ l a < e q for l n. (2.20 q For θ satisfying (2.20 we proceed in a different way. Write n e(ak θ = e(jθ ξ j n ( ( e j a ξj +. q q 0 (2.2 We replace ξ j by its expectation E[ξ j ] = j using again a random argument. Thus if n n ( ( e j a j n + (2.22 q q 0 we have log(2.2 log(2.22 ( ( ( ξ j E[ξ j ] log e j a +. (2.23 q q 0 Recall that q < R 3 (log J 3 (log n 3. Thus with high probability we may bound (2.23 by c n log log n log q < c n(log log n 3. Hence (2.2 e c n(log log n 3 (2.22.

10 0 J. BOURGAIN AND M.-C. CHANG Partition {,..., n} in intervals I = [rq, (r + q ] and estimate for each such interval j I q c q2 n q c q2 n ( ( e j a j n + q q 0 [ q 0 [ q 0 q s= ( ( e s a ] rq n + q q 0 q e s= ( s q ] rq n. (2.24 The product q ( s= e s q may be evaluated using Lemma 2. taking J = q 2, ρ =. q Thus clearly q ( s J log e q s= j J q j j < log q + C q cos 2πj s q + O( s= j + q j J q j j + O( implying that (2.24 < q c q2 n ( rq q+c n q 0elog < q c q2 n. (2.25 Since (2.22 is obtained as product of (2.24, (2.25 over the intervals I, we showed that (2.22 < q c q2 n n 2 q < e (log n3. Thus the preceding shows that if θ satisfies (2.20, then e(ak θ < e c n(log log n 3. (2.26 Going back to (2.0, omitting the case (2.20 estimated by (2.26, we obtained the bound cn(log log n 2 on (2.2 which permits to majorize (2.8 by c n log n(log log n and e(ak θ by e c nlog n log log n. This completes the proof of Proposition 2.2.

11 It was observed in [E-S] that ON A PAPER OF ERDÖS AND SZEKERES 3. Almost full proportion lim M(,...,n n (3. n exists and lies strictly between and 2. This fact is in contrast with Proposition 2.2 which gives a subset S {,...,n}, S n 2 s.t. However log M(S n(log n 2 log log n. (3.2 Proposition 3.. There is a constant τ > 0 such that if S {,...,n} satisfies S > ( τn, then for some c > 0. log M(S > cn (3.3 Thus (3.3 generalizes (3. in some sense, but in view of (3.2, it fails dramatically if we do not assume S small enough. n Proof of Proposition 3.. It will be convenient to use Fact 2 for an appropriate µ-convolution, which allow us to estimate the tail contribution in the k-summation. Thus consider min θ { j S = min θ k= k= j S cos 2πkj k ˆµ(jk k } µ (θ cos 2πkjθ min θ k 0 k= (log k 0 πn ˆµ(jk k k>k 0 ˆµ(jk k cos 2πkjθ (3.4 (3.5

12 2 J. BOURGAIN AND M.-C. CHANG since we assumed S > ( τn. Separating in (3.4 the cases k =, and 2 k k 0, we write ( (3.4 cos 2πjθ ˆµ(j k 0 k=2 k ˆµ(jk cos 2πkjθ. Take µ = F nr (θ, R > an appropriate constant and F nr (θ the Féjer kernel. (3.6 Thus F nr (s = s for s nr nr = 0 otherwise. Take θ = 3. The first term in (3.6 becomes, since 4n cosjx = 2 D n(x 2, where D n(x = sin(n + x 2 sin x 2 is the Dirichlet kernel, The second term is The third term becomes 2 sin 3π(n + 2n 2 2 sin 3π + 2 sin 3π. 4n 4n k 0 k k=2 j nr = n + 2R. ( jk nr + By partial summation, the inner sum is bounded by max j min(n, nr k j cosπ 3kj 2n ( 3 = max j min(n, nr k 2 D j 2 πk n 2 sin 3 4 π k n + 2. cosπ 3kj 2n 2. (3.7

13 For k < k 0 = o(n, the first term Hence It follows from the preceding that ON A PAPER OF ERDÖS AND SZEKERES 3 k 0 (3.7 k=2 2 sin 3π 4n ( sin 3π 4n for R a sufficiently large constant. = cn log k 0 2k sin 3π. 4n 2k 2 sin 3π 4n log k 0 ( π 2 6 log k 0. (2 π2 log k 0 n + 6 2R We bound (3.5 by (3.5 k k 0 k j nr k k k 0 nr k 2 R k 0 n. In summary, we proved that ˆµ(jk cos 2πjk 3 k 4n cn log k 0 τ(log k 0 n C n > c k 0 2 n j S k= be choosing first k 0 large enough and then assuming τ sufficiently small. This proves Proposition Sets with large arithmetical Diameter As we pointed out the general lower bound M(a,...,a n > n remains unimproved. However Proposition 4. stated below shows that in certain cases one can do better. First, we give the following definition. Definition. D = {v,...,v m } Z is called dissociated provided the relation ε v + + ε m v m = 0 with ε i = 0,, implies that ε = = ε m = 0.

14 4 J. BOURGAIN AND M.-C. CHANG We note that Hadamard lacunary sets are dissociated. Proposition 4.. Assume S = {a,...,a n } contains a dissociated set D of size m. Then log M(a,...,a n m 2 o(. (4. (log n 2 Thus (4. improves the general lower bound from [E-S] provided m > (log n 3+ε. Remark. By a result of Pisier [P], our assumption is equivalent to S containing a Sidon set Λ of size Λ m. Here Sidon set is in the harmonic analysis sense i.e. λ n e(nθ c λ n for all scalars {λ n } n Λ with c = c(λ to be considered as a constant. (This concept is different from the Sidon sets in combinatorics!. Dissociated sets are Sidon and conversely, Pisier proved that if Λ is a finite Sidon set, then Λ contains a proportional dissociated set. Proof of Proposition 4.. We derive (4. from the equivalent statement ( max log e(a θ + + log e(a n θ m 2 o( (4.2 θ (log n /2 which, since log e(aθ = 0 for a Z\{0}, is a consequence of the stronger claim that denoting Recall that by Fact with F m 2 o( (log n /2 (4.3 F(θ = log e(a θ + + log e(a n θ. F(θ = f(θ = k= f(kθ (4.4 k cos(2πa j θ.

15 ON A PAPER OF ERDÖS AND SZEKERES 5 We first perform a finite Mobius inversion on (4.4. Recall that if k = µ(d = 0 if < k r Hence d<r square free d k,d r d square free F(dθ µ(d d = = k= l= = f(θ d<r square free cos(2πa j dkθ µ(d dk [ cos(2πa j lθ ] µ(d l d l,d<r square free [ cos(2πa j lθ ] µ(d l l>r d l,d<r square free (4.5 where Note also that G(θ = = f(θ + G(θ, l>r d l,d<r square free where ω(l is the number of distinct prime factors of l. [ cos(2πa j lθ ] µ(d. l d l,d<r square free µ(d 2 ω(l, (4.6 Denote m the size of the largest dissociated set contained in {a,...,a n }. Our first task will be to bound the Fourier transform Ĝ of G. Thus given t Z, we have Ĝ(t 2 a t j t 2ω( a j. (4.7 We will bound (4.7 by considering dyadic ranges, letting for K > r dyadic J = J K = {j [, n] : a j t and t a j K}.

16 6 J. BOURGAIN AND M.-C. CHANG Thus j J a j t 2ω( t a j j J ( aj 2 ( 4 ω(k 2 t k K (4.8 ( J 2 K K J 2 (log K 2 2 = (log K 2. K Assume K J > (log K 8. (4.9 Our aim is to get a contradiction for appropriate choice of r. At this point, we invoke the following result from [H-T] (see Fq (.4. Denote ψ(x, y = {n x : if p n, then p y}. Lemma 4.2. For any 0 < α <, we have ψ ( x, (log x /α < x α+o( for x. (4.0 It follows from (4.9 that for any fixed > α > 0, we have We make the following construction. By (4., there is j J such that t a j = p b. J > 2ψ ( K, (log K α. (4. t a j has a prime divisor p > (log K α and we write Next, let J = {j J : p t a j }. Hence J < K K p + < (log K α assume α taken much smaller than. 8 < J (log K α 8 where we It follows that also J\J > ( 2 ψ ( K, (log K (log K α 8 α which permits to introduce j 2 J\J and a prime p 2 > (log K α t a j2 = p 2 b 2. Clearly p 2 p and p b 2. The contribution of the process is clear. We may introduce elements j,...,j s J with s (log K α 8 such that p 2 t a j2. Write

17 and prime divisors p s for s s and t a js ON A PAPER OF ERDÖS AND SZEKERES 7. Write t a js = p s b s such that p s t a js for s < s. Hence p s p s p s b s for s < s. (4.2 We claim that the set {a j,...,a js } is dissociated. Otherwise, there is a non-trivial relation ε a j + + ε s a js = 0 with ε s = 0,, which by the preceding translates in or ε p b + + ε s p s b s = 0 s s = ε s Let s be the smallest s with ε s 0. Then s Since identity (4.3 implies contradicting (4.2. s =s ε s p s s s p s b s = 0. p s b = 0. (4.3 s s s s s p s b s s s s s p s for s > s, s >s b s, Hence {a j,...,a js } is dissociated and by definition of m, s m implying m (log K α 8 and log r m α 8α. Thus, by taking log r m 2α (α small enough we obtain a contradiction under assumption (4.9.

18 8 J. BOURGAIN AND M.-C. CHANG Hence J K < K (log K 8 for K > r and summing (4.8 over dyadic ranges of K > r gives the bound Consequently Since Ĝ(t < (log K 2 log r. (4.4 K>r dyadic (4.5(t = ˆf(t + O( = log r ˆf(t + o( for all t Z. (4.5 ˆf(j = 2, we have (4.5(j = + o(. (4.6 2 Next, let D be a size m dissociated set in {a,...,a n }. Define ϕ(θ = e(jθ. m Also, let Φ, Ψ be the dual Orliez functions j D Φ(x = x log(2 + x and Ψ(x = e x2. It is well known (e.g. Theorem 3. in [Rud]. that By (4.6 It remains to bound (4.5 L Φ. ϕ L Ψ = O(. ( m 2 o( (4.5ϕ(θdθ C (4.5 L Φ (4.7 0 Estimate (4.5 log( ( dθ (4.8 2 j/2 µ(m dλ, j>0 2 2j λ 2 2j

19 ON A PAPER OF ERDÖS AND SZEKERES 9 Where M = {θ : (4.5(θ > λ} and µ is the measure. Using the left hand side of (4.5, the j-summands is bounded by Also, let Ψ (u = e u. Then d r since log e iθ L Ψ <. Thus also the bound 2 j/2 (4.5 2 j/2 log r F. (4.9 F(dθ d (log r F L Ψ n log r, L Ψ µ(m e c implying the following bound for the j-summands 2 j/2 2 2j e λ n log r j 22 c n log r. (4.20 Hence (4.8 < j 2 j/2 min ((log r F, 2 2j 22j c e n log r. For 2 2j 2 < n log r, we get the contribution For 2 2j 2 n log r, we bound by (log n 2 log r F. (n log r 4+ǫ e cnlog r + (n log r 4 2+ǫ e c(n(log r3 + + (n log r 4 2u +ǫ e c(n log r2u + < O(. Hence recalling above choice for log r. (4.5 L Φ (4.8 < (log n 2 m 2α F (4.2 Returning to (4.7, we proved that ( 2 o( m 2 2α (log n 2 F hence This proves (4.3 and hence Proposition 4.. F m 2 ε (log n 2.

20 20 J. BOURGAIN AND M.-C. CHANG Acknowledgment During the preparation of this paper, the first author was partially supported by the NSF Grant DMS 3069 and the second author by the NSF Grant DMS REFERENCES [A] F.V. Atkinson, On a problem of Erdos and Szekeres, Canad. Math. Bull., 4 (96, 7 2. [C] S. Chowla, The Riemann zeta and allied functions, Bull. Amer. Math. Soc, 58 (952, [C2] S. Chowla, Some applications of a method of A. Selberg, J. reine angew. Math., 27 (965, [E-S] P. Erdos, G. Szekeres, On the product n k= ( za k, Publ. de l Institut mathematique, 950. [K-O] Y. Katznelson, D. Ornstein, The differentiability of the conjugation of certain diffeomorphisms to the circle, ETDA 9 (989, no 4, [Kol] M.N. Kolountzakis, On nonnegative cosine polynomials with nonnegative integral coefficients, Proc. AMS 20 (994, [Kol2] M.N. Kolountzakis, A construction related to the cosine problem, Proc. Amer. Math. Soc. 22 (994, vol. 4, 5 9. [Kol3] M.N. Kolountzakis, Some applications of probability to additive number theory and harmonic analysis, Number theory (New York Seminar, ,Springer, New York, (996, [Kol4] M.N. Kolountzakis, The density of B h [g] sets and the minimum of dense cosine sums, J. Number Theory 56 (996,, 4-. [H-T] A. Hildebrand, G. Tenenbaum, Integers without large prime factors, J. Theorie des Nombres de Bordeaux, 5 (993, no 2, [O] A.M. Odlyzko, Minima of cosine sums and maxima of polynomials on the unit circle, J. London Math. Soc (2, 26 (982, no 3, [P] G. Pisier, Arithmetic Characterization of Sidon Sets, 8 (983, Bull. AMS, [Rud] W. Rudin, Trigonometric series with gaps, J. Math. Mech., 9 (960, [R] I.Z. Rusza, Negative values of cosine sums, Acta Arith. (2004, [S] A. Schinzel, On the number of irreducible factors of a polynomial, Topics in number theory (ed. P. Turan, North Holland, Amsterdam, (976, INSTITUTE FOR ADVANCED STUDY, PRINCETON, NJ address: bourgain@math.ias.edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, RIVERSIDE, CA address: mcc@math.ucr.edu