An Addition Theorem Modulo p

Similar documents
Week 15-16: Combinatorial Design

Example. Addition, subtraction, and multiplication in Z are binary operations; division in Z is not ( Z).

Solutions to Assignment 1

SUMSETS MOD p ØYSTEIN J. RØDSETH

Constructing c-ary Perfect Factors

NUMBER SYSTEMS. Number theory is the study of the integers. We denote the set of integers by Z:

Journal of Number Theory

RMT 2013 Power Round Solutions February 2, 2013

Subset sums modulo a prime

Sums of Squares. Bianca Homberg and Minna Liu

FUNCTIONS OVER THE RESIDUE FIELD MODULO A PRIME. Introduction

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Weak Uniform Distribution for Divisor Functions. I

AN EXTENSION OF A THEOREM OF EULER. 1. Introduction

Arithmetic progressions in sumsets

SMT 2013 Power Round Solutions February 2, 2013

PRACTICE PROBLEMS: SET 1


Number Theory Homework.

Use mathematical induction in Exercises 3 17 to prove summation formulae. Be sure to identify where you use the inductive hypothesis.

Fadwa S. Abu Muriefah. ON THE DIOPHANTINE EQUATION x fc = y n

Subrings and Ideals 2.1 INTRODUCTION 2.2 SUBRING

On Systems of Diagonal Forms II

Acta Arith., 140(2009), no. 4, ON BIALOSTOCKI S CONJECTURE FOR ZERO-SUM SEQUENCES. 1. Introduction

Combinatorial Batch Codes and Transversal Matroids

Math 324 Summer 2012 Elementary Number Theory Notes on Mathematical Induction

Notes on Systems of Linear Congruences

OQ4867. Let ABC be a triangle and AA 1 BB 1 CC 1 = {M} where A 1 BC, B 1 CA, C 1 AB. Determine all points M for which ana 1...

THE DENSITY OF THE PRODUCT OF ARITHMETIC PROGRESSION

SUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM MOD P

(1) N(bi, n) < cid ;-

Solutions to Assignment 3

arxiv: v1 [math.co] 18 May 2018

Maximum exponent of boolean circulant matrices with constant number of nonzero entries in its generating vector

Problems and Results in Additive Combinatorics

Some zero-sum constants with weights

Mathematics 228(Q1), Assignment 3 Solutions

*!5(/i,*)=t(-i)*-, fty.

Solving a linear equation in a set of integers II

A PROBLEM IN ADDITIVE NUMBER THEORY*

Maximal Number of Subsets of a Finite Set No k of Which are Pairwise Disjoint

1 Lecture 1 (1/5/2009)

1 Lecture 1 (1/5/2009)

Wilson s Theorem and Fermat s Little Theorem

THREE-BIT MONOMIAL HYPEROVALS IN PG ( 2, 2 h)

. As the binomial coefficients are integers we have that. 2 n(n 1).

SUM-PRODUCT ESTIMATES APPLIED TO WARING S PROBLEM OVER FINITE FIELDS

Know the Well-ordering principle: Any set of positive integers which has at least one element contains a smallest element.

Arithmetic Progressions with Constant Weight

MULTIPLICITIES OF MONOMIAL IDEALS

arxiv: v1 [math.ac] 7 Feb 2009

A diametric theorem for edges. R. Ahlswede and L.H. Khachatrian

The edge-density for K 2,t minors

ON THE DENSITY OF SOME SEQUENCES OF INTEGERS P. ERDOS

An Exploration of the Arithmetic Derivative

ON A CUBIC CONGRUENCE IN THREE VARIABLES. IP

MATH 324 Summer 2011 Elementary Number Theory. Notes on Mathematical Induction. Recall the following axiom for the set of integers.

64 P. ERDOS AND E. G. STRAUS It is possible that (1.1) has only a finite number of solutions for fixed t and variable k > 2, but we cannot prove this

Algebraic number theory

Appendix C Vector and matrix algebra

Rings, Modules, and Linear Algebra. Sean Sather-Wagstaff

= 1 2x. x 2 a ) 0 (mod p n ), (x 2 + 2a + a2. x a ) 2

GENERALIZATIONS OF SOME ZERO-SUM THEOREMS. Sukumar Das Adhikari Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad , INDIA

On Locating-Dominating Codes in Binary Hamming Spaces

Basic Proof Examples

Minimal Decompositions of Hypergraphs into Mutually Isomorphic Subhypergraphs

SOLUTIONS TO PROBLEM SET 1. Section = 2 3, 1. n n + 1. k(k + 1) k=1 k(k + 1) + 1 (n + 1)(n + 2) n + 2,

ON ADDITIVE PARTITIONS OF SETS OF POSITIVE INTEGERS

A misère-play -operator

Arithmetic Properties for a Certain Family of Knot Diagrams

ARCS IN FINITE PROJECTIVE SPACES. Basic objects and definitions

Maximal Class Numbers of CM Number Fields

BULLETIN DE L'ACADEMIE POLONAISE DES SCIENCES Série des sci. math., astr. et phys. - Vol. VI, No. 12, 1958 On Sets Which Are Measured bar Multiples of

We want to show P (n) is true for all integers

On a Class of (0, 1) Matrices with Vanishing Determinants*

2013 University of New South Wales School Mathematics Competition

1 Take-home exam and final exam study guide

Chapter 4. Matrices and Matrix Rings

4.1 Primary Decompositions

The primitive root theorem

Small Sets Which Meet All the k(n)-term Arithmetic Progressions in the Interval [1;n]

AND OCCUPANCY PROBLEMS

Groups with many subnormal subgroups. *

Wilson s theorem for block designs Talks given at LSBU June August 2014 Tony Forbes

252 P. ERDÖS [December sequence of integers then for some m, g(m) >_ 1. Theorem 1 would follow from u,(n) = 0(n/(logn) 1/2 ). THEOREM 2. u 2 <<(n) < c

Math 3000 Section 003 Intro to Abstract Math Homework 6

2-Distance Problems. Combinatorics, 2016 Fall, USTC Week 16, Dec 20&22. Theorem 1. (Frankl-Wilson, 1981) If F is an L-intersecting family in 2 [n],

Neighbor-distinguishing k-tuple edge-colorings of graphs

The Distribution of Generalized Sum-of-Digits Functions in Residue Classes

4 a b 1 1 c 1 d 3 e 2 f g 6 h i j k 7 l m n o 3 p q 5 r 2 s 4 t 3 3 u v 2

(V/í) : Y, A) - 0 mod 2. (V-tf) : Y,a) = 0 mod 2.

Number Theory Solutions Packet

Discrete Mathematics, Spring 2004 Homework 4 Sample Solutions

MATH 215 Final. M4. For all a, b in Z, a b = b a.

J. Combin. Theory Ser. A 116(2009), no. 8, A NEW EXTENSION OF THE ERDŐS-HEILBRONN CONJECTURE

INTEGERS REPRESENTED BY POSITIVE TERNARY QUADRATIC FORMS* 1. Introduction. Dirichletf proved by the method of 2 the following two theorems:

On prime factors of subset sums

SOME AMAZING PROPERTIES OF THE FUNCTION f(x) = x 2 * David M. Goldschmidt University of California, Berkeley U.S.A.

PERIODICITY OF SOME RECURRENCE SEQUENCES MODULO M

Chapter 3 Basic Number Theory

Transcription:

JOURNAL OF COMBINATORIAL THEORY 5, 45-52 (1968) An Addition Theorem Modulo p JOHN E. OLSON The University of Wisconsin, Madison, Wisconsin 53706 Communicated by Gian-Carlo Rota ABSTRACT Let al,..., a= be distinct non-zero residue classes modulo a prime p. In this paper we estimate the number of residue classes represented by qa~ -4- "'" q- E=a= where the E~ are restricted to the values 0 and 1. In particular, we verify a conjecture of P. Erd6s and H. Heilbronn: every residue class is represented if s > 2p 1/2. 1. INTRODUCTION Let al,..., a= be distinct non-zero residue classes modulo the prime p and let r be the number of residue classes x of the form Jr = eial -q- --" Jc e=as, (1) where the Ei are restricted to the values 0 and 1 and are not all 0. P. ErdSs and H. Heilbronn [1] showed that r = p if s > 3(6p) 1/2 and conjectured that r =pifs>2p ~/2. In this paper we prove THEOREM 1. If S > (4p -- 3)1/2, then r = p. As shown in [1], this is nearly best possible: If = = ( 1 s-11s al 1, a2 --1... a= = -- )" [~( + I)] and s < (4p + 5) 1/2 -- 2, then (for p > 3) the residue 89 + 1) cannot be expressed in the form (1). We shall obtain Theorem 1 as an easy consequence of THEOREM 2. Let ai,..., a= be non-zero residue classes modulo p such that ai :;& for i J= j, and let p be the number of residue classes (including O) of the form qal + "'" + %a=, Ei = 0 or 1. 45

46 OLSON /f then l s2+s<~p+ l, 2;" + 3s ~ 2p + 5, s ~ 0 (mod 2) s ~- 1 (mod 2), (2) o>~l+-- s(s -I- 1) (3) And in any case I min l ~ -+-3 1 + s(s + 1) I ~'s~-o(mod2) P >/ 2 ' 2 (4) tminl p+32 's(s+l) I2 ~'s~ l(mod2). The proof of Theorem 2 is elementary and is based on ideas contained in [1]. 2. PROOF OF THEOREM 1 Let G denote the additive group of residue classes modulo p. For subsets A, B of G, let A B be the set of all sums a + b, aea, b EB, let L A I be the number of elements in A, and let A be the complement of A in G. We give the proof of Theorem 1 in the case s = 3 (mod 4); the proofs for the other three cases, where we obtain slightly smaller lower estimates for s, are similar. Put s--1 s+l u-- 2,v= 2 ' and arrange the notation so that as =/= --aj if either 1 ~< i <j u+l ~<i<j~<s. Let ~< u or S = {0, at} + "'" + {0, au}, T = {0, au+l} + "'" + {0, a~}. By Theorem 2 SI /> min IP TI >/min I p / +2 3., u(u 2 + 1) t >/z'----w---p + 1 +32,1+ v(v-{-1) 1= P + 2 1

AN ADDITION THEOREM MODULO p 47 Let T' be the set T with zero removed. It follows that I S + T' I = P, since IS] +1 T'[ ~>p+ 1. 3. PROOF OF THEOREM 2 We first prove three lemmas. Let B be a non-empty subset of G. Define a mapping AB from G into the non-negative integers by AB(x) = I(x q- B) n/? l, x ~ a. Thus AB(x) is the number of representations ofx as a difference x = b -- b, b ~ B, l) ~.B. LEMMA 2.1. (i) (ii) (iii) (iv) PROOF: (i) (ii) ;t(o) = O. Let A = AB, where B is a set of size I B I = k. A(-x) = A(x), x ~ G. A(x + y) <~ A(x) + A(y), x, y E G. If C is a subset of G of size ] C I = t and O ~ C, then Clear. A(--x)= (iii) A(x + y) ~" A(c) >~ k(t -- k 1). e~c l(-x + B) n B I k--[(--x+b)ab] k-- I gn (g + x) l a(x). = [(x+y+b) n/2r = [(y + B) n (--x +/?)l = [(y q- B) n (--x q-/2) n B [ + I(y + B) n (--x + B) n B I ~< [(--x+/])nbt +[(y+b) n/~] = I B n (x q- B)f + I(Y B) n/~ 1 = a(x) -5 A(y). (iv) c~c [(c + B) nb[ <~ ~[(x-+ B)nB] =k(k. 1). c~g x=/=o

48 OLSON Thus ccc h(c) = ~ [k -- [(c + B) ~ B I] >/ kt--k(k-- 1) = k(t--k4-1). cec LEMMA 2.2. Let A 1... A~ be subsets of G of the same size [ Ai [ = m > 1, none of which is in arithmetic progression, and such that 0 ~ Ai and --Ai=Ai. Then ] A1 + "'" + Ar I ~> min{p,r(m + 1) -- 1}. Pnoov: The lemma is a simple consequence of a theorem of A. G. Vosper [2, Th. 1.3, p. 3]. The lemma is clear for r = 1. Assume r > 1 and that the lemma holds for r-- 1. Set C =AI+... +At. Vosper's theorem gives I C[ ~> min(p -- 1, i AI + "" 4- At_ 1 [ @ ] A r l} ~> min(p -- 1, r(m + 1) -- 2}. But m is odd and, since 0 ~ C and --C = C, [ C [ is odd. Thus / C[ ~> min{p,r(m + 1) -- 1}. LEMMA 2.3. Let A and B be subsets of G of sizes [ A [ = n, [ B [ = k. Assume 0 (~ A, --A = A, and A w {0} is not in arithmetic progression. Let t be an integer, 1 <~ t <~ p -- 1, and set Set )t = An and Then t =r(n4-2) +q (--1 <~ q <~ n). c~ = max A(a), a ~ A. and 2(n 4-2)k(t -- k Jr- 1) c~ ~ t(t4-n4-6)+q(n--q--2)' (5) 8(n4-2) k(t--k+ 1) >~ 4t(t4-n+6)+(n--2) 2 " (6) PROOF: Equation (6) follows from (5) since q(n value k(n -- 2) 2. q -- 2) has maximal

AN ADDITION THEOREM MODULO 19 49 We construct a subset C of G with 0 r C of size I C [ = t such that I t(t+n+6)+q(n--q--2) A(c) ~ c~ 2(n + 2) " c~c I (7) Equation (5) then follows from (7) and statement (iv) of Lemma 2.1. If t ~ n and therefore r = 0, q = t, let C consist of t of the elements in A. Thus ~],~c A(e) ~ at and (7) holds. Assume that t ~ n + 1 and therefore that r i~ 1. Let A * = A u {0}, and let Cj = A* +.." + A*. j times By Lemma 2.2, Cj contains at least min{p -- 1,j(n + 2) -- 2} non-zero elements. Thus we may form a set C of non-zero elements of size t which contains at least j(n + 2) -- 2 elements from each C~ (1 ~< j ~< r) and at most q + 2 elements from Cr+~ c~ Cr. Now if c E C~-, c :?6 0, then ' a' c : al+"'+ v where a~... a'~ are (not necessarily distinct) elements of A and 1 ~< v ~< j. Hence h(c) ~ ~ A(a~) ~ vo~ ~ j~, by Lemma 2.1. It follows that A(c) ~<na+(n+2) 2a+'"+(n+2)ra+(q+2)(r+ 1)o~ c~c ~ {r(r + 1)(n + 2)+ 2(q + 2)(r + 1) -- 4} 2{(r+ 1)(t+q+4)--4} This proves (7), and hence (5). t t(t+ n +6) + g(n q-- 2) 2(n + 2) ~" We now use Lemma 2.3 and statement (ii) of Lemma 2.1 to prove Theorem 2. Let al..., as be non-zero elements of G such that ai ~ ~a~ for i ~-j. Set B = B(al... a3 = {0, al) +... + {0, as}, and let A be the set of 2s elements..., 582/5/~-4

50 OLSON CASE 1: A U {0} is in arithmetic progression. We may assume without loss of generality that A u {0} is in progression with difference. 1. Therefore, rearranging subscripts, ai ~ +1 (modp), i = 1... s. Ifxa, x2... x, is any sequence of elements in G, then clearly Let f B(al... a~)[ = {x1, a I + x1} dr_... 27 {xs, as q- x~)l. (8) Equation (8) becomes Clearly l 0 if ai~i(modp) xi = --ai if ai ~ --i (mod p). I B J = I{0, 1} +... + {0, s}l. [B]: rain lp, 1 -}-s(s q-2 l)t which proves Theorem 2 in this ease. CASE 2: A W {0} is not in arithmetic progression. Set A :AB, and ~ =max)~(ai) :maxa(a), a~a. Arrange the notation so that A(a~) = ~. If ~ 1, put B* = B(al... a~-o. We show first that IB >~ I B* I -{-~. (9) If b ~ B and a, + b ~ B, then clearly b q~ B*. Hence ]BI~>IB*I +l(a~+b) nbi :lb*lq-c~. We prove the first part of Theorem 2 by induction on s. Assume s > 1, s satisfies (2), and We show next that ]B*I >~ 1 -k (s-- 1)s (10) 2 [ B[ >~ s(s q- 1) (11) 2

AN ADDITION THEOREM MODULO p 51 Set n = I A ] = 2s, k = I B I, and t = 2k -- 2 in (6). We may assume that s(s + 1) IBI~ 2 ' and hence t ~< p -- 1 by (2). By (6) and the inequality k>~ 1 +(s-- 1)s 2 we have ~ > s -- 2. Therefore ~ ~ s -- 1. Equation (11) now follows from (9) and (10). To prove (3) it suffices to show that ~ > s -- 1. We obtain this from (5) by setting and therefore n =2s, k--s(s+ 1) 2 ' I s -t- s -- 2 if s ~ 0 (mod 2) t ~ 3s -- 7 s2+ 2 ifs~ l(mod2), 2s q = 3s -- 5 9 if s ~ 0 (mod 2) ifs ~ 1 (mod 2). Statement (4) holds if s satisfies (2). Let So be the smallest positive integer which fails to satisfy (2). We are done if F B(al a,o)t >! p + 3 '"" 2 Put s = s o and assume [ B] ~ (p -]- 1)/2. By (3) we have ]B*[ >/ 1 q- (S~ 1)So 2 Set n ~ 2s, t = 2k -- 2 in (6). Using k >/1 q- (So-- 1)So 2 ' we have o~ > So -- 2. Therefore o~/> So -- 1 and I B(al a,o) I ~ So(So -+- 1) '" 2

52 OLSON If so ~ 0 (mod 2), we have So(So q-1)~<p q-1, contradicting the significance of So 9 Hence we must have ] B(ax... a,%)l ~ p + 3 2 ' which proves (4), If s0 ~ 1 (mod 2), we have shown that (4) holds for s = So. In this case it suffices to show Deny (12). Thus B(aa as)[ ~ p 3 where s = So + 1. (12) '" 2 ' k=fbi ~P +1 k~s(s--l ~) 9 2 ' 2 By setting t = 2k -- 2, n = 2s in (6) we get ~ > s -- 3. Hence B(al,..., as)l >is(s--2 1) q-s--2 F------f--P-}- 3 This completes the proof of Theorem 2. REFERENCES 1. P. ERD6S AND H. HEILBRONN, On the Addition of Residue Classes mod p, Acta Arith. 9 (1964), 149-159. 2. H. B. MANN, Addition Theorems, Wiley, New York, 1965.

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback