Különböző részösszegű aszimptotikus bázisok Fizika, Földtudományok és Matematika

35 OTDK, Fizika, Földtudományok és Matematika Szekció, Algebra és számelmélet Tagozat.

Különböző részösszegű aszimptotikus bázisok

Hallgató: Nguyen Vinh Hung
Szak: Matematika, Képzés típusa: bsc, Intézmény: Budapesti Műszaki és Gazdaságtudományi Egyetem, Kar: Természettudományi Kar

Témavazető: Dr. Kiss Sándor - egyetemi docens, Budapesti Műszaki és Gazdaságtudományi Egyetem Természettudományi Kar

A set of natural numbers is called asymptotic basis of order k if every large enough natural number can be written as the sum of k terms from the set. A set of natural numbers is called generalized g-Sidon set if every natural number can be written as the sum of h terms from the set at most g times. Over many years, the generalized g-Sidon sets which are asymptotic bases of some order were investigated by many authors. According to a famous conjecture of Erdős and Turán, an asymptotic basis of order 2 cannot be a B_2[g] set. Later, P. Erdős , A. Sárközy and V. T. Sós asked if there exists a Sidon set which is an asymptotic basis of order 3. G.Grekos, L. Haddad, C.Helou and J.Pihko showed that a Sidon set cannot be an asymptotic basis of order 2. J. M. Deshouillers and A. Plagne introduced a construction for a Sidon set which is an asymptotic basis of order at most 7. The existence of Sidon sets which are asymptotic bases of order 5 was also proved with the help of probabilistic tools by Sándor Kiss. In addition, there is an improvement: it was showed that there exists Sidon sets that are asymptotic bases of order 4. Also, it was proved the existence of generalized 2-Sidon sets which are an asymptotic bases of order 3. Furthermore, in 1985, Erdős asked if there exists an asymptotic basis of order k with the property that all the sums formed by at most k-1 terms are all distinct. We prove the existence of an asymptotic basis of order 2k+3 such that all the k terms sums from this asymptotic basis are all different. In the proof we use probabilistic and combinatorial methods.