On a question of Cochrane and Pinner concerning multiplicative subgroups
aa r X i v : . [ m a t h . N T ] M a y On a question of Cochrane and Pinner concerningmultiplicative subgroups
Tomasz Schoen ∗ and Ilya D. Shkredov † Abstract
Answering a question of Cochrane and Pinner, we prove that for any ε > , suffi-ciently large prime number p and an arbitrary multiplicative subgroup R of the field F ∗ p , p ε ≤ | R | ≤ p / − ε the following holds | R ± R | ≥ | R | + δ , where δ > depends on ε only.
1. Introduction.
Let p be a prime number, and R ⊆ F ∗ p be a multiplicative subgroup. Such subgroupswere studied by various authors, see e.g. [1]–[10], [17], [19]. Heath–Brown and Konyagin[8] proved that for any multiplicative subgroup R ⊆ F ∗ p with | R | = O ( p / ) we have | R ± R | ≫ | R | / . Obviously, the result is best possible for subgroups of size approximately p / . On the other hand Cochrane and Pinner asked about the possibility of improvingthe last bound for smaller subgroups (see [5], Question 2). The aim of this paper is toanswer their question in the affirmative. Our main result can be stated as follows. Theorem 1.1
Let ε ∈ (0 , be a real number. Then there exists a positive inte-ger p ( ε ) and a real δ ∈ (0 , , δ = δ ( ε ) such that for all primes p ≥ p and everymultiplicative subgroup R ⊆ F ∗ p , p ε ≤ | R | ≤ cp / the following holds | R ± R | ≫ min {| R | p (log | R | ) − , | R | p δ } . (1) Furthermore, for every | R | ≤ cp / , we have | R ± R | ≫ min (cid:8) | R | p (log | R | ) − , max {| R | p − (log | R | ) − , | R | p − (log | R | ) − } (cid:9) . (2)The formula (1) can be applied for small subgroups and the inequality (2) gives goodbounds for large subgroups. In particular | R | / p − / (log | R | ) − / ≫ | R | / , providedthat | R | > p / ε , ε > | R | ∼ p / then (2) yields | R ± R | ≫ | R | / − ε . In [6] Glibichuk proved the following interesting result (see also [13]). ∗ The author is partially supported by MNSW grant N N201 543538. † The author is supported Pierre Deligne’s grant based on his 2004 Balzan prize, President’s of RussianFederation grant N –1959.2009.1, grant RFFI N 06-01-00383 and grant Leading Scientific Schools No.691.2008.1 heorem 1.2 Let
A, B ⊆ F p be two sets, | A || B | > p . Suppose that B = − B or B ∩ ( − B ) = ∅ . Then AB = F p . Corollary 1.3
Let R ⊆ F ∗ p be a multiplicative subgroup and | R | > √ p . Then R = F p . We derive from Theorem 1.1 that for all sufficiently large p , in Corollary 1.3 onecan replace 8 R by 6 R (for precise formulation see Theorem 4.1 below). The last resultis connected with an intriguing question concerning basis properties of multiplicativesubgroups. Let R ⊆ F ∗ p be such a subgroup, | R | ≥ p / ε , ε >
0. What is the least l such that lR contains F ∗ p ? A well–known hypothesis states that l equals two. As wasshowed in [19] the last hypothesis is not true in general finite fields F p n , n → ∞ even forsubgroups R with restriction | R | ≤ | F p n | / − ε , ε > A be a subset of an abeliangroup G . A common argument, which can be found in many proofs of additive resultsuses the estimate of energy E ( A ) ≥ | A | / | S | , where S = A − A. We show, roughlyspeaking, that if E ( A ) does not exceed | A | − ε / | S | (but in a stronger sense, actually weuse higher moments of convolution, see Corollary 3.2) then typically | A − A s | ≫ | S | − cε , where A s = A ∩ ( A − s ). Then, using an obvious inequality | S ∩ ( S − s ) | ≥ | A − A s | (3)we prove that E ( S ) ≫ | S | − c ′ ε . Now, assume that A is a multiplicative subgroup satisfying | S | ≪ | A | / ε (the case | A + A | ≪ | A | / ε is very similar). From a result of Heath–Brown and Konyagin [8]it follows immediately that E ( A ) ≪ | A | ε / | S | . Therefore, we have E ( S ) ≫ | S | − c ′ ε . However, the last inequality cannot be true provided that ε is small enough. It is easyto observe that the set S just a union of some cosets and we know that each of thecoset is uniformly distributed (see Theorem 2.7 and Corollary 2.5 below). The obtainedcontradiction completes the proof.We conclude with few comments regarding the notation used in this paper. Alllogarithms used in the paper are to base 2 . By ≪ and ≫ we denote the usual Vinogradov’ssymbols. Finally, with a slight abuse of notation we use the same letter to denote a set S ⊆ G and its characteristic function S : G → { , } . The second author is grateful to S.V. Konyagin for useful discussions.
2. Previous results.
In this section we collect basic definitions we shall use later on and quickly recallknown additive properties of multiplicative subgroups.Let G be a finite Abelian group, N = | G | . It is well–known [12] that the dual group b G is isomorphic to G . Let f be a function from G to C . We denote the Fourier transformof f by b f , b f ( ξ ) = X x ∈ G f ( x ) e ( − ξ · x ) , (4)2here e ( x ) = e πix . We rely on the following basic identities X x ∈ G | f ( x ) | = 1 N X ξ ∈ b G (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) . (5) X y ∈ G (cid:12)(cid:12)(cid:12) X x ∈ G f ( x ) g ( y − x ) (cid:12)(cid:12)(cid:12) = 1 N X ξ ∈ b G (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) (cid:12)(cid:12)b g ( ξ ) (cid:12)(cid:12) . (6)If ( f ∗ g )( x ) := X y ∈ G f ( y ) g ( x − y ) and ( f ◦ g )( x ) := X y ∈ G f ( y ) g ( y + x )then [ f ∗ g = b f b g and [ f ◦ g = b f b g . (7)Let also f c ( x ) := f ( − x ) for any function f : G → C .Write E ( A, B ) for additive energy of two sets
A, B ⊆ G (see e.g. [18]), that is E ( A, B ) = |{ a + b = a + b : a , a ∈ A, b , b ∈ B }| . If A = B we simply write E ( A ) instead of E ( A, A ) . Clearly, E ( A, B ) = X x ( A ∗ B )( x ) = X x ( A ◦ B )( x ) = X x ( A ◦ A )( x )( B ◦ B )( x ) , (8)and by (6), E ( A, B ) = 1 N X ξ | b A ( ξ ) | | b B ( ξ ) | . (9)Now let G = F p , where p is a prime number. We call a set Q ⊆ F p R –invariant if QR = Q. First of all let us estimate Fourier coefficients of an arbitrary R –invariant set. Lemma 2.1
Let R ⊆ F ∗ p be a multiplicative subgroup and let Q be a nonempty R –invariant set. Then for all ξ = 0 the following holds | b Q ( ξ ) | < | Q | / p / | R | − / . (10) Proof.
By Parseval identity and R –invariance | R || b Q ( ξ ) | ≤ X ξ =0 | b Q ( ξ ) | = p | Q | − | Q | < p | Q | and the result follows. (cid:3) Using Stepanov’s method, Heath-Brown and Konyagin proved the following theorem(see [8]).
Theorem 2.2
Let R ⊆ F ∗ p be a multiplicative subgroup and let Q ⊆ F ∗ p be a R –invariant set such that that | Q | ≪ p | R | . Then X ξ ∈ Q ( R ◦ R )( ξ ) ≪ | R || Q | / . (11)3e shall apply a lemma from [8] (see also [11, 10]) which is a consequence of thetheorem above. Lemma 2.3
Let R ⊆ F ∗ p be a multiplicative subgroup, | R | = O ( p / ) . Then E ( R ) ≪ | R | / . By Cauchy–Schwarz inequality it follows immediately that | R ± R | ≫ | R | / . Weshall use another consequence of Lemma 2.3. Lemma 2.4
Let R ⊆ F ∗ p be a multiplicative subgroup, | R | = O ( p / ) , and let Q be anonempty R –invariant set. Then for all ξ = 0 the following holds | b Q ( ξ ) | ≪ | Q | / p / | R | − / . (12) Proof.
We use the fact that Q is a disjoint union of some cosets x j R , j =1 , , . . . , | Q | / | R | . By H¨older inequality, R –invariance and Lemma 2.3, we have | b Q ( ξ ) | ≤ | Q | / | R | X i =1 | b R ( x i ξ ) | ≤ (cid:16) | Q | / | R | X i =1 | b R ( x i ξ ) | (cid:17) / ( | Q | / | R | ) / = | Q | / | R | − (cid:16) | Q | / | R | X i =1 | R || b R ( x i ξ ) | (cid:17) / = | Q | / | R | − (cid:16) X x | b R ( x ) | (cid:17) / ≤ | Q | / | R | − ( p E ( R )) / ≪ | Q | / p / | R | − / and the result follows. (cid:3) Also Lemma 2.1, Lemma 2.4 and Lemma 2.3 implies the following upper estimatefor Fourier coefficients of a multiplicative subgroup (see [11] or [10]). For completenesswe recall the proof.
Corollary 2.5
Let R ⊆ F ∗ p be a multiplicative subgroup. Then max ξ =0 | b R ( ξ ) | < √ p .Suppose, in addition, that | R | = O ( p / ) . Then max ξ =0 | b R ( ξ ) | ≪ min { p | R | , p | R | } . (13) Proof.
Let ρ = max x =0 | b R ( x ) | . The first estimate ρ < √ p is a consequence onLemma 2.1. We should check (13). To obtain the first bound in the formula, we applyLemma 2.4 with Q = R . Further, let ξ = 0 be an arbitrary residual. By R –invarianceand H¨older inequality we have | R | | b R ( ξ ) | = (cid:12)(cid:12)(cid:12) X x,y ∈ R e ( − ξxy ) (cid:12)(cid:12)(cid:12) ≤ | R | X x ∈ R | b R ( ξx ) | = | R | X x ( ξR ∗ ( ξR ) c )( x ) c R c ( x ) . | R | | b R ( ξ ) | ≤ | R | X x ( ξR ∗ ( ξR ) c )( x ) · X x ( ξR ∗ ( ξR ) c )( x ) | b R ( x ) | = | R | X ξ ( ξR ∗ ( ξR ) c )( ξ ) | b R ( x ) | , and | R | | b R ( ξ ) | ≤ | R | E ( R ) p . Now the assertion follows from Lemma 2.3. (cid:3)
From the above estimates of Fourier coefficients of subgroups one can deduce its basisproperties.
Theorem 2.6
Let R ⊆ F ∗ p be a multiplicative subgroup, l ≥ be a positive integer.Suppose that | R | ≫ min { p l +25 l − , p l +53 l +3 } . Then F ∗ p ⊆ lR . If l ≥ and | R | > p l +12 l then F ∗ p ⊆ lR . Proof.
Suppose that lR + F ∗ p . Then for some λ = 0 we have0 = X x b R l ( x ) c λR ( x ) = | R | l +1 + X x =0 b R l ( x ) c λR ( x ) . (14)Using Corollary 2.5 and Parseval formula to estimate the second term in (14), we getthe required result. (cid:3) In particular, if l = 7 and | R | ≫ √ p then F ∗ p ⊆ lR . For large l better bounds areknown (see [11, 3]).Finally, we recall a well–known result of Bourgain, Glibichuk and Konyagin [3] onFourier coefficients of multiplicative subgroups. Theorem 2.7
Let ε ∈ (0 , be a real number. Then there exists a positive integer p ( ε ) and a real η ∈ (0 , such that for all primes p ≥ p and any multiplicative subgroup R ⊆ F ∗ p , | R | ≥ p ε we have max λ ∈ F ∗ p (cid:12)(cid:12)(cid:12) X x ∈ R e ( λx ) (cid:12)(cid:12)(cid:12) ≤ | R | p − η . (15)Notice that for large subgroups (i.e. | R | > p / ε , ε >
0) better estimates hold (seeCorollary 2.5 above and [8, 11, 10]). Finally, we remark that the condition | R | ≥ p ε inthe theorem above can be replaced by a weaker inequality | R | ≫ exp( C ′ log p/ log log p ),where C ′ >
3. Additive combinatorics.
We return for a moment to a general case of an arbitrary Abelian group G . For anyset A ⊆ G and any element s ∈ A − A define the set A s = A ∩ ( A − s ) (see papers[9]—[16]). Clearly, A s = ∅ and | A s | = ( A ◦ A )( − s ) = ( A ◦ A )( s ). Furthermore, set E ( A ) = X s ( A ◦ A )( s ) E ( A ) = X s ( A ◦ A )( s ) . The following simple lemma exhibits interesting relations between energies of A s and thequantities E ( A ) , E ( A ) . Lemma 3.1
Let G be an Abelian group. For every set A ⊆ G we have X s ∈ A − A E ( A, A s ) = E ( A ) and X s ∈ A − A X t ∈ A − A E ( A s , A t ) = E ( A ) . Proof.
Observe that for every x, w, s ∈ G A s ( x ) A s ( x + t ) = A ( x ) A ( x + s ) A ( x + t ) A ( x + s + t ) = A t ( x ) A t ( x + s ) . Summing over x , we get( A s ◦ A s )( t ) = ( A t ◦ A t )( s ) , s, t ∈ G . Clearly, X s ( A t ◦ A t )( s ) = | A t | = ( A ◦ A )( t ) . Thus, by (3.1) we have X s E ( A, A s ) = X s X t ( A ◦ A )( t )( A s ◦ A s )( t ) = X t ( A ◦ A )( t ) X s ( A t ◦ A t )( s ) = E ( A ) . Similarly, X s X t E ( A s , A t ) = X s X t X x ( A s ◦ A s )( x )( A t ◦ A t )( x )= X x X s ( A x ◦ A x )( s ) X t ( A x ◦ A x )( t ) = E ( A ) , which proves Lemma 3.1. (cid:3) Next, we show that if E ( A ) and E ( A ) are small then typically | A − A s | and | A s − A t | are large, respectively. Corollary 3.2
Let A be a subset of an abelian group G with at least elements.Then X s =0 | A ± A s | ≥ − | A | E ( A ) − (16) and X s =0 X t =0 | A s ± A t | ≥ − | A | E ( A ) − . (17)6 n particular there exist s = 0 such that | A − A s | ≥ − | A | E ( A ) − | A − A | − and s, t = 0 such that | A s − A t | ≥ − | A | E ( A ) − | A − A | − . Proof.
Fix s ∈ A − A . By Cauchy–Schwarz inequality and (8), we obtain( | A || A s | ) = (cid:16) X z ( A ◦ A s )( z ) (cid:17) = (cid:16) X z ( A ∗ A s )( z ) (cid:17) ≤ E ( A, A s ) · | A ± A s | , so that | A | ( | A | − | A | ) = X s ∈ ( A − A ) \{ } | A || A s | ≤ X s ∈ ( A − A ) \{ } E ( A, A s ) / · | A ± A s | / . Applying once again Cauchy–Schwarz inequality and Lemma 3.1, we have2 − | A | ≤ X s =0 E ( A, A s ) / | A ± A s | / ≤ (cid:16) X s =0 E ( A, A s ) (cid:17) / (cid:16) X s =0 | A ± A s | (cid:17) / ≤ E ( A ) / (cid:16) X s =0 | A ± A s | (cid:17) / . The second inequality one can prove using very similar argument. (cid:3)
We bound from the above E ( R ) and E ( R ), where R is a multiplicative subgroup. Lemma 3.3
Let R ⊆ F ∗ p be a multiplicative subgroup and | R | ≪ p / . Then E ( R ) ≪ | R | log | R | and E ( R ) ≤ | R | + O ( | R | / ) . (18) Proof.
Let a be a parameter, and put n = | R | . We have E ( R ) ≤ a n + X s : ( A ◦ A )( s ) ≥ a ( A ◦ A )( s ) . Let us arrange values ( A ◦ A )( s ), s ∈ F p /R in decreasing order and denote its values as N ≥ N ≥ . . . . By Theorem 2.2, we have N j ≪ n / j − / . Hence E ( R ) ≤ a n + n X j : j ≪ n /a N j ≪ a n + n X j : j ≪ n /a j − ≪ a n + n log n a . Taking a = n / we obtain the required result. To establish the second estimate, onesimilarly bounds the contribution of terms with s = 0 to E ( R ) by ≪ | R | / . (cid:3) Remark.
Suppose that R is a multiplicative subgroup, | R | ≪ p / . Clearly, by (9)we have E ( A ) ≥ p − | A | for every set A ⊆ G . Hence, by Corollary 3.2 and Lemma3.3, we can find s ∈ R − R , s = 0 such that | R ± R s | ≫ | R | p log | R | . It gives interestingconsequences for subgroups R satisfying | R | ≥ cp / , where c > s ∈ R − R , s = 0 with | R ± R s | ≫ p log | R | . Actually, from(16) one can deduce that there are at least ≫ | R | p log | R | ∼ p log | R | such s . Notice that therestriction | R | ≥ cp / implies | R − R | ≫ p (see e.g. [17]). Furthermore, from Corollary7.2 and Lemma 3.3 it follows immediately that for some s, t = 0 we have | R s ± R | ≫ p / . However, in this case we can prove even more. Let P ⊆ ( R − R ) \ { } be the set ofall popular differences, i.e. s ∈ F ∗ p having at least | R | / (10 p ) ≫ p / representations as r − r , r , r ∈ R . By H¨older inequality, (17) and (18) we have | R | ≪ X s,t ∈ P | R s || R t | ≤ (cid:0) X s,t ∈ P E ( R s , R t ) (cid:1) / (cid:0) X s,t ∈ P ( | R s || R t | ) (cid:1) / (cid:16) X s,t ∈ P | R s ± R t || R s || R t | (cid:17) / ≤ E ( R ) / E ( R ) / (cid:16) X s,t ∈ P | R s ± R t || R s || R t | (cid:17) / ≤ | R | / (cid:16) X s,t ∈ P | R s ± R t || R s || R t | (cid:17) / , hence | R | ≪ X s,t ∈ P | R s ± R t || R s || R t | . Thus, if | R | ≫ p / there are ≫ p pairs s, t such that | R s | , | R t | ≫ p / and | R s − R t | ≫| R s || R t | .
4. Proof of Theorem 1.1.
Let S = R − R , S ′ = R + R , n = | R | , and m = | S | . We may assume that n ≥ S \{ } and S ′ \{ } are R –invariant. Now we apply arguments used[9], [14]—[16] Fix s ∈ S . In view of ( A − A s ) ⊆ S , ( A − A s ) ⊆ S − s and ( A + A s ) ⊆ S ′ ,( A + A s ) ⊆ S ′ − s , we have( S ◦ S )( s ) ≥ | A − A s | , ( S ′ ◦ S ′ )( s ) ≥ | A + A s | , s ∈ S .
We will deal with the sign ”-” and the set S . The case of S ′ can be handle in the sameway. By Corollary 3.2 and (4) n E ( R ) − ≪ X s S ( s )( S ◦ S )( s ) . Using Fourier transform and Parseval formula, we infer that n E ( R ) − p ≪ X ξ | b S ( ξ ) | b S ( ξ ) ≤ m + ρmp , where ρ = max ξ =0 | b S ( ξ ) | . Whence either n E ( R ) − p ≪ m or n E ( R ) − ≪ ρm . In thefirst case, by Lemma 3.3, we have np / (log n ) − / ≪ m . Now suppose that n E ( R ) − ≪ ρm . By Theorem 2.7 ρ ≤ mp − η . Therefore, by Lemma3.3, we get n (log n ) − ≪ m p − η , which proves (1). Here one can take any δ < η/ n E ( R ) − ≪ ρm . Using Lemma2.1, we have ρ < ( pm/n ) / + 1. Hence n (log n ) − ≪ m / p / n − / ,
8o that m ≫ n / p − / (log n ) − / . Finally, applying Lemma 2.4, we obtain n (log n ) − ≪ m / p / n − / , which gives m ≫ n / p − / (log n ) − / . This completes the proof. (cid:3) Our main result allows us to refine Theorem 2.6.
Theorem 4.1
Let R ⊆ F ∗ p be a multiplicative subgroup and κ > . Suppose that | R | ≥ p κ , then for all sufficiently large p we have F ∗ p ⊆ R . Proof.
Suppose that R ⊆ F ∗ p is a multiplicative subgroup with n = | R | ≥ p κ , κ > .Furthermore, put S = R + R , n = | R | , m = | S | , and ρ = max ξ =0 | b R ( ξ ) | . If 6 R F ∗ p thenfor some λ = 0 , we have0 = X ξ b S ( ξ ) b R ( ξ ) c λR ( ξ ) = m n + X ξ =0 b S ( ξ ) b R ( ξ ) c λR ( ξ ) . Therefore, by the second inequality of (13) and Parseval identity we get n m ≤ ρ mp ≪ ( p / n / ) mp. Now applying the second inequality of (2), m ≫ n / p − / (log n ) − / , we obtain therequired result. (cid:3) Finally, let us remark that using similar argument as in the proof of Theorem 4.1one can show that | R | ≥ p/ , provided that | R | ≥ p κ , where κ > / , and p is largeenough. References [1]
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