Terence Tao s harmonic analysis method on restricted sumsets

Transcription

1 Huaiyin Normal University Terence Tao s harmonic analysis method on restricted sumsets Guo Song

2 0. Abstract Let p be a prime, and A, B be finite subsets of Z p. Set A + B = {a + b : a A, b B}. Cauchy-Davenport theorem asserts that A + B min{p, A + B 1}. In 2005, Terence Tao gave a harmonic analysis proof of the Cauchy- Davenport theorem, by applying a new form of the uncertainty principle on Fourier transform. Recently Guo song and Sun zhi-wei gave some applications of Tao s harmonic analysis method to restricted sumsets. In this article we will introduce the progress in this field

3 1. Restricted sumsets Let p be a prime, and A, B be finite subsets of Z p. Set and A + B = {a + b : a A, b B} (1) A +B = {a + b : a A, b B, a b}. (2) Cauchy-Davenport theorem asserts that A + B min{p, A + B 1}. (3)

4 A well-known result on restricted sumsets states that A +A min{p, 2 A 3}, (4) which is conjectured by P.Erdős and H.Heilbronn [6] in 1964 and confirmed by J.A.Dias da Silva and Y.O.Hamidoune [5] in In N.Alon, M.B.Nathanson and I.Z.Ruzsa[2] proposed a polynomial method in this field and showed that if B > A > 0 then A +B min{p, A + B 2}. (5) By the powerful polynomial method(cf.[1],[2],[3], [8], [9],[11]), mathematicians gave many interesting results

5 2. Uncertainty principle Difinition 1 Let G be a locally compact Abel group and T be the unit circle group on complex plane. Let ν : G T be a continuous group homomorphism(such that ν(x) = 1 and ν(x + y) = ν(x)ν(y)). The group called dual group of G. Ĝ of all ν is Difinition 2 The Fourier transform on G is a function F T : L 1 (G) C 0 (Ĝ), f ˆf such that ˆf(ν) = ν( x)f(x)dx, ν Ĝ, where dx denote the Haar measure on G.

6 Let G be the field Z p. For f : Z p C is any complex-valued function, we may define its support supp(f) and its Fourier transform ˆf : Z p C as follow: supp(f) = {x Z p : f(x) 0} (6) and ˆf(x) = a Zp f(a)e p (ax), x Z p. (7) where e p (y) = e 2πiy/p for y Z p.

7 Tao obtained the following result in [12] : Theorem 1 Let p be an odd prime. If f : Z p C is not identically zero, then supp(f) + supp(ˆf) p + 1. (8) Given two non-empty subsets A and B of Z p with A + B p + 1, we can find a function f : Z p C with supp(f) = A and supp(ˆf) = B. Note that the inequality was also discovered independently by Andráas Biró

8 3. Tao s proof of Cauchy-Davenport theorem Let m = max{p + 2 A B, 1}, there exist subsets X, Y of Z p such that X = p + 1 A, Y = p + 1 B and X Y = m. By Theorem 1, there exist functions f, g such that supp(f) = A, supp( ˆf) = X, supp(g) = B, supp(ĝ) = Y

9 Let Observe that F (x) = f g(x) = 1 f(a)g(x a). p a Z p and thus we have supp(f ) supp(f) + supp(g) = A + B supp( ˆF ) supp( ˆf) + supp(ĝ) = X Y, A + B p + 1 X Y = min(p, A + B 1).

10 4. Variant of Tao s method with application to restricted sumsets A pair (Â, ˆB) of subsets of Z p is said to be m-good if 0  and p m ˆB but we don t have both p m + t  and p t ˆB for all t [0, m 1]. Let (Â, ˆB) m = {x Z p : x + t  and x + m t ˆB for some t [0, m]}.

11 Theorem 2 Let A and B be non-empty subsets of Z p with p odd prime, and C = {a + b : a A, b B, a b S} (9) with S Z p and S = m. Let Â, ˆB be subsets of Z p with  p + 1 A and ˆB p + 1 B. If (Â, ˆB) is m-good, then C p + 1 (Â, ˆB) m

12 Proof of Theorem 2: By Theorem 1 there are functions f, g : Z p C such that supp(f) = A, supp(ˆf) = Â, supp(g) = B and supp(ĝ) = ˆB. Now we define a function F : Z p C by F (x) = f(a)g(x a) p (x a) e p (a d)). (10) a Z p d S(e For each x supp(f), there exists a supp(f) with x a supp(g) and d := a (x a) S, hence x = a + (x a) C. Therefore supp(f) C. (11)

13 For any x Z we have ˆF (x) = b Zp where F (b)e p (bx) = a Z p P (a, b) = (e p (b a) e p (a d)) = d S T S f(a)g(b a)e p (bx)p (a, b), b Z p ( 1) T ( e p (( S T )(b a))e p T a d ). d T

14 Therefore ˆF (x) = ( 1) T ( e p d ) f(a)e p (ax+ T a) g(b a)e p ((b a)x+( S T )(b a)) T S d T a Z p b Z p = ( 1) T ( e p d ) ˆf(x + T )ĝ(x + m T ). T S d T By the definition of m-good pair we have so ˆF is not identically zero. ˆF (p m) = ( 1) m e p ( d S d) ˆf(0)ĝ(p m) 0,

15 Suppose that x supp(ˆf). Then there is a subset T of S with T = t such that x + t  = supp( ˆf) and x + m t ˆB = supp(ĝ), hence x (Â, ˆB) m. Thus supp(ˆf) (Â, ˆB) m. With the help of Theorem 1, we have C supp(f) p + 1 supp(ˆf) p + 1 (Â, ˆB) m. This concludes the proof

16 Construct suitable m-good pair (Â, ˆB) so that [Â, ˆB] m is small and hence C is large. Example 1 Let  = { 0, 1,..., A 1} and ˆB = {p S B + 1, p S B + 2,..., p S }, then (Â, ˆB) is m-good.

17 Hence, Theorem 3 Let A and B be non-empty subsets of Z p with p odd prime, and C = {a + b : a A, b B, a b S} (12) with S Z p. Then we have C min{p, A + B 2 S 1}. (13)

18 Example 2 Suppose that m+1 2 A A + B p m. Set k = p+1 A, l = p + 1 B and n = k l. Suppose that [2 m] n m 2. Let  = {2i : i = 0, 1,..., k 1} and ˆB = {x : x [1, 2k 1 p]} {x : x p m(mod 2) & x [2k p, p+1+2l 2k]} Then (Â, ˆB) is m-good with (Â, ˆB) m 2k + m p

19 With the help of some m-good pairs, we can obtain that Theorem 4 Let A and B be non-empty subsets of Z p with p odd prime, and C = {a + b : a A, b B, a b S} (14) with S Z p. Then we have C min{p, A + B S r}, (15) where r = 2 if A = B and S 1(mod 2), r = 1 + min{ S 2, A B } otherwise.

20 In [7] S. Guo and Z.Sun conjecture that min{ S 2, A B } can reduced, hence r = 2 if A = B and S 1(mod 2), r = 1 otherwise. When S is even, the conjecture was proposed by Q.Hou and Z.Sun in [8]. In the case S Z p, H. Pan and Z. Sun [11, Corollary 2] obtained the inequality C min{p, A + B S 2} (16) via the polynomial method.

21 References [1] N.Alon, Combinatorial Nullstellensatz, Combin. Prob. Comput., 8, (1999), [2] N.Alon, M.B.Nathanson and I.Z.Ruzsa, The polynomial method and restricted sums of congruence classes, J. Number Theory, 56, (1996), [3] H.Q.Cao and Z.W.Sun, On sums of distinct representatives, Acta Arith., 87, (1998), [4] H.Davenport, On the addition of residue classes, J. London Math. Soc., 10, (1935),

22 References [5] J.A.Dias da Silva and Y.O.Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc., 26, (1994), [6] P.Erdős and H.Heilbronn, On the addition of residue classes modulo p, Acta Arith., 9, (1964), [7] S. Guo and Z.W.Sun, A variant of Tao s method with application to restricted sumsets, J.Number Theory, 129(2009); [8] Q.H.Hou and Z.W.Sun, Restricted sums in a field, Acta Arith., 102, (2002),

23 References [9] J.X.Liu and Z.W.Sun, Sums of subsets with polynomial restrictions, J. Number Theory, 97, (2002), [10] M.B.Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets (Graduated texts in mathematics; 165), Springer, New York, [11] H.Pan and Z.W.Sun, A lower bound for {a + b : a A, b B, P (a, b) 0}, J. Combin. Theory Ser. A, 100, (2002), [12] Terence Tao, An uncertainty principle for cyclic groups of prime order, Math. Res. Lett., 12, (2005),

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