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Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A note on a problem of Hilliker and Straus. | A note on a problem of Hilliker and Straus Miroslawa Janczak Faculty of Mathematics and CS Adam Mickiewicz University ul. Umultowska 87 61-614 Poznan Poland mjanczak@amu.edu.pl Submitted May 20 2006 Accepted Oct 23 2007 Published Oct 30 2007 Mathematics Subject Classifications 06124 06124 Abstract For a prime p and a vector ã ã1 . ak 2 Zp let f ã p be the largest n such that in each set A c Zp of n elements one can find x which has a unique representation in the form x ã1a1 ãkak di 2 A. Hilliker and Straus 2 bounded f ã p from below by an expression which contained the L1-norm of ã and asked if there exists a positive constant c k so that f ã p c k log p. In this note we answer their question in the affirmative and show that for large k one can take c k O 1 k log 2k . We also give a lower bound for the size of a set A c Zp such that every element of A A has at least K representations in the form a a a a 2 A. 1 Introduction Let f p denote the largest number n such that in any set A a1 . ang contained in Zp ZỊpZ at least one difference di dj is incongruent to all other differences. Straus 4 estimated f p up to a constant factor showing that 1 log2 p - 1 1 f p itA1 log2 p 2 log2 3 for all primes p. Hilliker and Straus 2 studied the following natural generalization of the problem. For a given vector ã a1 . ak E Zp consider the set of all linear combinations S S ã A ã1 A a2A akA. Let f ã p be the largest n such that for any set A c Zp IAI n one can find at least one element which has the unique representation in S. They proved that ft- log p _ b I 1 f ã p lAtii-ii 1 M2INI1 THE ELECTRONIC JOURNAL OF COMBINATORICS 14 2007 N23 1 where ã 1 P 1 ãị . They ask if the L1 -norm of a vector ã can be replaced by a function which depends only on k i.e. if f ã p c k log p In the note we settle the above problem in the affirmative Theorem 1 Corollary 1 below . We also show that our lower bound for f ã p given in Theorem 1 cannot be much improved Theorem 2 . In section 3 we find a .
