SUMSETS MOD p ØYSTEIN J. RØDSETH

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1 SUMSETS MOD p ØYSTEIN J. RØDSETH Abstract. In this paper we present some basic addition theorems modulo a prime p and look at various proof techniques. We open with the Cauchy-Davenport theorem and a proof using the Davenport transform. We continue with a result, due to Davenport, on strongly connected digraphs. We include a brief application of Philip Hall s theorem on distinct representatives to sumsets mod p. Then we give a simple proof of Vosper s theorem using the Davenport transform. Finally, we look at sums of distinct residue classes mod p. Complete results for this type of restricted sumsets were first obtained using representation theory and multilinear algebra. These results do not seem to lend themselves to transformation proofs. However, now we have the beautiful technique called the polynomial method, which gives simple proofs of such results. We demonstrate the polynomial method by using it in a second proof of the Cauchy-Davenport theorem. 1. Introduction Let A be a set of k 1 distinct integers a 1, a 2,...,a k, and let B be a set of l 1 distinct integers b 1, b 2,...,b l. The sumset A + B is defined as A + B = {a + b a A, b B}. (1) Thus the sumset A + B consists of the distinct integers appearing in the rectangular array a 1 + b 1 a 1 + b 2... a 1 + b l a 2 + b 1. a 2 + b a 2 + b l. (2) a k + b 1 a k + b 2... a k + b l What can be said about the number A + B of elements in A + B? That is, what can be said about the number of distinct integers in the array (2)? Trivially, we have A+B kl, and it might very well happen that all the integers in the array (2) actually are distinct. What about a lower bound for A + B? We arrange the notation such that a 1 < a 2 < < a k and b 1 < b 2 < < b l. In the array (2) we start at the upper left hand corner, move along the first row, and continue down the last column. Then we have a strictly increasing sequence of integers in A + B, a 1 + b 1, a 1 + b 2,...,a 1 + b l, a 2 + b l,..., a k + b l. (3) 2000 Mathematics Subject Classification. 11A07. Key words and phrases. Sumsets, congruences, residue classes mod p. 1

2 2 Ø. J. RØDSETH This sequence consists of k + l 1 distinct integers. Hence, A + B k + l 1. (4) By choosing A and B to be arithmetic progressions with a common difference, we see that this result is best possible. Let p be a prime. What happens to the considerations above if we replace the integers by residue classes modulo p? Thus A = {a 1, a 2,...,a k } and B = {b 1, b 2,...,b l } are now subsets of Z/pZ, the finite field of p elements. We continue to write k for the number of elements in A, and the number of elements in B is l. The sumset A + B is defined by (1). Consequently, A + B consists of the distinct residue classes among those listed in the array (2). We continue to write A + B for the number of elements in A + B. Still we have, of course, that A + B kl, or rather A + B min(p, kl). The question is if (4) still holds. The answer is not unconditionally yes. We always have A + B p, so if p < k + l 1, then the answer is no. But let us assume that k +l 1 p. Is (4) then correct? The proof above for ordinary integers, where we looked at the sequence (3), fails for residue classes. It is true that the residue classes in the first row in (2) are all distinct, and the same goes for the residue classes in the last column. But more than one of the elements in the first row may very well also appear in the last column. Nevertheless, the inequality (4) does hold. Theorem 1 (Cauchy-Davenport). A + B min(p, k + l 1). This is a basic result in the part of additive/combinatorial number theory known as addition theorems mod p (and more generally as addition theorems in groups ). The theorem was proved by Cauchy [4] in 1813 and rediscovered by Davenport [6, 7] in As we have already indicated, the proof is more complicated than that for integers. Both Cauchy and Davenport employed what are known as transformation proofs. In the literature it is mainly the so-called (Dyson) e-transform that is used. This is essentially the same transform as the one employed by Cauchy. The Davenport transform is more or less forgotten. In this paper we use the Davenport transform to prove Theorem 1. This does not offer special advantages over the use of the e-transform in the proof of Theorem 1. But in Section 7 we shall see that the Davenport transform gives a rather nice proof of Vosper s theorem. This paper is organized in 8 sections. In Section 2 we introduce some notation. In Section 3 we describe the Davenport transform. Then, in a brief Section 4 we prove Theorem 1. In Section 5 we show a result about directed graphs, a result which, according to Davenport, is equivalent to Theorem 1. In Section 6 we employ Philip Hall s theorem on distinct representatives to say a bit more about where in the array (2) we can find k +l 1 distinct residue classes when k +l 1 p. In Section 7 we consider an inverse theorem mod p, and in Section 8 we close by mentioning a type of result which does not seem to lend itself to any transformation proof.

3 SUMSETS MOD p 3 2. Notation As before, let A and B be non-empty subsets of Z/pZ. The sumset A + B is defined by (1). This set operation is commutative and associative. Also the sum of more than two sets is uniquely defined. In particular, we write ha for A + + A (h addends). For c Z/pZ we write c+a for {c}+a. We set B = { b b B}, and A B = A+( B). The number of elements in A is denoted by A. We write A B when A is a subset of B. We reserve the symbol for proper inclusion. Thus A B means that A is contained in B, but is not equal to B. We denote the relative complement of B in A by A \ B, while B denotes the complement of B in Z/pZ. If there are residue classes a, d Z/pZ such that A consists of the distinct residue classes a, a + d, a + 2d,..., a + (k 1)d, then A is an arithmetic progression with difference d. For an integer r we will on some occasions also write r for the residue class modulo p represented by r. 3. The Davenport Transform We now introduce the Davenport transform (in modern notation), and give its most important properties. Suppose that 0 B, B 2, and A + B Z/pZ. Then A + B A + 2B. If A + B = A + 2B, then A + B = A + 2B = A + 3B =... For an a A and 0 b B, we then have a + nb A + B for n = 0, 1, 2,... Thus we have A + B = Z/pZ, contrary to assumptions. Therefore A + B A + 2B. Putting we thus have X. For x X we set X = (A + 2B) \ (A + B), (5) B x = {b B x b A + B}, B x = B \ B x. We shall call B x for a Davenport transform of B. Now, we have 0 B x, and 0 B x B. Clearly, We also have (A + B x ) (x B x) A + B. (6) (A + B x ) (x B x) =, (7) since we (in self-explanatory notation) have that if a+b x = x b x, then x b x = a+b x A + B, so that b x Bx, a contradiction. By (6) and (7) we thus have A + B A + B x + x B x, that is, A + B A + B x + B B x. (8)

4 4 Ø. J. RØDSETH 4. Proof of Theorem 1 Assume the theorem false. Then there are pairs of sets A, B such that A + B Z/pZ, and A + B A + B 2. (9) These properties remain valid if we replace B by b+b for some b B. We may therefore assume that 0 B. Choose such a pair A, B for which B is minimal. Then B 2. By Section 3, there exists a Davenport transform B x such that (8) holds. We have 1 B x < B and A + B x < A + B < p. By (8) and (9), A + B x A + B B + B x A + B x 2, which contradicts the minimality of B. 5. Digraphs The Cauchy-Davenport theorem is given as statement A in Davenport s 1935 paper. What seems to be less known, is that Davenport s paper also contains a statement B, which he describes as equivalent to statement A. By this, he obviously means that statement B follows easily from statement A, and vice versa. Davenport did not use modern graph-theoretic language. Today we can formulate Davenport s statement B as follows: Let B, V Z/pZ, where 0 B. Set E = {(x, y) V V y x B}. Then D = (V, E) is a digraph with vertex set V and edge set E. For a moment, think of V as fixed, while B is allowed to vary. If B is small, then there are few edges, and the digraph D falls apart into several pieces (the graph D is disconnected). On the other side, if B is large, then we probably have many edges, and the chances should be good of D being in one piece (the graph D is connected). If B is large, the chances should even be good of D being strongly connected, that is, there is a directed path from any vertex v V to any other vertex w V. Therefore the following theorem, which is Davenport s statement B, appears quite natural. Theorem 2 (Davenport). If B > p V, then D is strongly connected. Let us deduce this theorem from the Cauchy-Davenport theorem. Let v V, and By putting B 0 = {0} B, we have A = {a V v = a or there is a path from v to a}. (A + B 0 ) V A. To see this, let a + b 0 = w (A + B 0 ) V. If b 0 = 0, then w = a A. If b 0 B, we have w a = b 0 B, such that there is an edge from a A to w. Hence w A. So, we have A (A + B 0 ) V A, that is, (A + B 0 ) V = A.

5 By inclusion-exclusion, we get SUMSETS MOD p 5 p (A + B 0 ) V = A + B 0 + V (A + B 0 ) V, that is, p V A + B 0 A. We now assume that B > p V. Then B > A + B 0 A, or since B 0 = B + 1, A + B 0 < A + B 0 1. The Cauchy-Davenport Theorem now gives A +B 0 = p, that is A + B 0 = Z/pZ. Hence, A = (A + B 0 ) V = (Z/pZ) V = V, which means that there is a path from an arbitrary vertex v to any other vertex. This completes the proof of Theorem Distinct Representatives Davenport s paper [6] begins on page 30 in the 1935 volume of Journal of the London Mathematical Society. On this very page ends a paper by Philip Hall [11]. Hall s paper contains his famous theorem about distinct representatives. Theorem 3 (P. Hall). Let C 1, C 2,..., C k be sets. Suppose that the union of every selection of s of these sets contains at least s elements for all s = 1, 2,..., k. Then we can find a set of k distinct elements, one from each of the k sets. There are many proofs of this result in the literature. Our two favourites are the ones presented in [1, p. 28] (due to D. J. Shoesmith) and [19, p. 116] (due to R. Rado). As before, let A = {a 1,...,a k }, B = {b 1,...,b l } Z/pZ, A = k 1, B = l 1. Suppose that A + B Z/pZ. By the Cauchy-Davenport theorem we have A + B k + l 1. All a i +b j are given by (2). Is it possible to say something about where in the rectangular array (2) we can find k + l 1 distinct residue classes? Let C be an (l 1)-subset of A + B. (For example, C = a 1 + {b 1,...,b l 1 }.) Then we may choose one element from each row in (2), such that C and these elements constitute k + l 1 distinct residue classes in (2). (We may, for instance, choose all elements in the first row in (2) and one element from each of the other rows.) This is seen in the following way. The set of elements in row i in the rectangular array (2) is equal to a i + B. Let C i = (a i + B) \ C. For 1 i 1 < i 2 < < i s k we have, by the Cauchy-Davenport theorem, C i1... C is = ({a i1,...,a is } + B) \ C {a i1,...,a is } + B C s + l 1 (l 1) = s,

6 6 Ø. J. RØDSETH and by Hall s theorem we can find k distinct elements, one from each C i. Along with the l 1 elements of C, we thus have k + l 1 distinct elements of A + B. 7. An Inverse Theorem mod p The Cauchy-Davenport Theorem is a direct addition theorem mod p. Given sets A and B, such a theorem usually gives a result about A + B; in our case a lower bound for A + B. The corresponding inverse problem is to describe the structure of the sets A, B for which A + B is small. The first non-trivial inverse theorem mod p is due to Vosper [17]. Theorem 4 (Vosper). We have A + B = min(p, A + B 1) if and only if one of the following conditions is satisfied: (i) A + B > p, (ii) A = 1 or B = 1, (iii) B = c A for a c Z/pZ, (iv) A and B are arithmetic progressions with the same difference. The principal step in the proof of Vosper s theorem is the proof of the following result. Theorem 5 (Vosper). Suppose that B 2, and that Then A is an arithmetic progression. A + B = A + B 1 < p 1. (10) Vosper first used the Davenport transform to prove this result. Later he gave in [18] a simpler proof by employing the e-transform. Another transform was employed by Chowla, Mann, and Straus in [5], where they also gave a beautiful application of Vosper s theorem to diagonal forms over Z/pZ; cf. [15, p. 57], [14, Ch. 2]. Now we shall show how the Davenport transform gives us an even simpler proof of Theorem 5 than Vosper s second. Proof of Theorem 5. Assume that the theorem is false. Let A, B be a pair of sets such that A does not form an arithmetic progression, (10) holds, and B 2 is minimal. We may also assume that 0 B. Let X be given by (5). For an x X, let B x be the corresponding Davenport transform of B. By (8) and (10), we then have A + B x A + B B + B x A + B x 1 < p 2. By the minimality of B we have B x = {0}. Setting B = B \ {0}, we thus have B x = B for all x X, so that x B A + B for all x X, and we see that A (X B ) A + B and A (X B ) =.

7 It follows by (10) that and application of Theorem 1 gives SUMSETS MOD p 7 A + B 1 = A + B A + X B, B 1 X B X + B 1 = X + B 2. Thus we have X 1, that is, X = 1. Since A + B < p 1, it follows that Applying Theorem 1 once more, we obtain A + 2B = A + B + 1 < p. 1 = A + 2B A + B A + B + B 1 A + B = B 1. Hence B = 2. If 0 b B, the conditions upon A and B also hold for the sets {ab 1 a A} and {0, 1}. Therefore it is no restriction to assume that B = {0, 1}. Thus we have by (10), A + {0, 1} = A + 1. (11) Consider the residue classes 0, 1,..., p 1 as consecutive points on a circle. By (11), we have exactly one element a A with a + 1 A. Hence, the elements of A form a set of consecutive points on the circle, that is, A is an arithmetic progression with difference 1. This completes the proof of Theorem 5. If we in Theorem 5 also assume that A 2, it is now easy to see that B is an arithmetic progression with the same difference as A. For we may assume that A = {0, 1,..., A 1}. Then A = {0, 1}+A for A = {0, 1,..., A 2}. By (10) we have A+B < p, and using Theorem 1 we get that is, A + B 1 = A + B = {0, 1} + A + B {0, 1} + B + A 1 = {0, 1} + B + A 2; {0, 1} + B B + 1. (12) If we once more regard the residue classes mod p as points on a circle, we see that (12) holds if and only if B is an arithmetic progression with difference 1. The proof of Theorem 4 is now easily completed. In [12] the authors go one step beyond Vosper, and in [13] another step is taken. But we are far away from existing conjectures generalizing Vosper s theorem. On the other hand, related to Vosper s theorem, there is a strong inverse theorem by Freiman [10, Theorem 2.1], [15, Theorem 2.11]. This theorem of Freiman is the subject of [16].

8 8 Ø. J. RØDSETH 8. The Polynomial Method We define another type of addition of sets of residue classes mod p by putting As before, we let A = k and B = l. By the Cauchy-Davenport theorem we have A +B = {a + b a A, b B, a b}. (13) A + A min(p, 2k 1). More than 40 years ago, Erdős and Heilbronn conjectured that we have the similar result A +A min(p, 2k 3). (14) If (14) is true, it is best possible. This is seen by taking A as an arithmetic progression. If we want to prove (14) by transformation, we actually have to prove a more general result for the set (13), since one or both sets are altered by transformation. But this plan does not work, since the condition a b in (13) is destroyed by all (known) transformations. So a non-transformation proof is what we need. Then it is natural first to try to find such a proof of the Cauchy-Davenport theorem. And the first non-transformation proof of the Cauchy-Davenport theorem was given in 1990 by Dias da Silva and Hamidoune [8] using multilinear algebra. In 1994, more than 30 years after Erdős and Heilbronn stated their conjecture, Dias da Silva and Hamidoune [9] proved the conjecture using results from representation theory and multilinear algebra. Soon after, Alon, Nathanson, and Ruzsa [2, 3] found a beautiful elementary proof of (14). They even proved that A +B min(p, k + l 2} if k l. (15) Alon, Nathanson, and Ruzsa s new method of proof is called the polynomial method. To demonstrate this method, we now use this technique to give a second proof of the Cauchy-Davenport theorem. We consider polynomials over the field Z/pZ. We put f(x, y) = (x + y c), and define g(x, y) by c A+B g(x, y) f(x, y) (mod a A (x a), b B(y b)), where g(x, y) is of degree at most k 1 in x, and of degree at most l 1 in y. We have g(a, b) = f(a, b) = 0 for all (a, b) A B. A well-known theorem of Lagrange states that if a polynomial of degree at most n 1 in one variable over Z/pZ has n zeros, then the polynomial is identically zero. It is now a simple exercise to show that g(x, y) is the zero polynomial.

9 We have f(x, y) = A+B j=0 ( A + B j SUMSETS MOD p 9 ) x j y A+B j + terms of lower degree. Suppose that A + B p. Then the monomial x k 1 y A+B k+1 has non-zero coefficient, and the exponent of y must be reducible when the monomial is taken mod b B (y b). Thus A + B k + 1 l, and the proof is complete. To prove (15), consider the polynomial h(x, y) = (x y) (x + y c) instead of f(x, y). c Ab+B While it is straightforward to generalize the Cauchy-Davenport theorem to the sum of more than two sets, this is not the case with (14) or (15). Dias da Silva and Hamidoune nevertheless showed how (14) generalizes to h equal addends, while Alon, Nathanson, and Ruzsa showed how (15) generalizes to h not necessarily equal sets. The polynomial method has its obvious advantages to transformation methods. But the opposite is also true. It is an unsolved problem to produce inverse results by the polynomial method. Nobody has succeeded in determining the structure of those A for which equality holds in (14), nor the structure of those A, B for which equality holds in (15). Nor do we know if the polynomial method can be used to prove Vosper s theorem. Nevertheless, problems of the type touched upon in this paper constitute a very active field of research. References [1] I. Anderson, A First Course in Combinatorial Mathematics, Second ed., Oxford University Press, New York, [2] N. Alon, M. B. Nathanson, and I. Z. Ruzsa, Adding distinct congruence classes modulo a prime, American Math. Monthly 102 (1995), [3] N. Alon, M. B. Nathanson, and I. Z. Ruzsa, The polynomial method and restricted sums of residue classes, J. Number Theory 56 (1996), [4] A. L. Cauchy, Recherches sur les nombres, J. École Polytech. 9 (1813), ; also in Oevres, Série 2, Tome 1, [5] S. Chowla, H. B. Mann, and E. G. Straus, Some applications of the Cauchy-Davenport theorem, Det Kongelige Norske Videnskabers Selskabs Skrifter 32 (1959), [6] H. Davenport, On the addition of residue classes, J. London Math. Soc. 10 (1935), [7] H. Davenport, A historical note, J. London Math. Soc. 22 (1947), [8] J. A. Dias da Silva and Y. O. Hamidoune, A note on the minimal polynomial of the Kronecker sum of two linear operators, Linear Algebra and its Applications 141 (1990), [9] J. A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmann derivatives and additive theory, Bull. London Math. Soc. 26 (1994), [10] G. A. Freiman, Foundations of a Structural Theory of Set Addition, Translations of Mathematical Monographs, Vol. 37, American Math. Soc., Providence, R. I. (1973).

10 10 Ø. J. RØDSETH [11] P. Hall, On representatives of subsets, J. London Math. Soc. 10 (1935), [12] Y. O. Hamidoune and Ø. J. Rødseth, An inverse theorem mod p, Acta Arith., 92 (3) (2000), [13] Y. O. Hamidoune, O. Serra, and G. Zémor, On the critical pair theory in Z/pZ, Acta Arith. 121 (2) (2006), [14] H. B. Mann, Addition Theorems: The Addition Theorems of Group Theory and Number Theory, Interscience Publishers, New York [15] M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets, Springer-Verlag, New York [16] Ø. J. Rødseth, On Freiman s 2.4-theorem, Trans. R. Norw. Soc. Sci. Lett. (to appear). [17] A. G. Vosper, The critical pairs of subsets of a group of prime order, J. London Math. Soc. 31 (1956), [18] A. G. Vosper, Addendum to The critical pairs of subsets of a group of prime order, J. London Math. Soc. 31, (1956), [19] R. J. Wilson, Introduction to Graph Theory, Forth ed., Longman, Essex Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway address: Oystein.Rodseth@uib.no URL: