WORKING WITH CHRISTIAN MAUDUIT (In memoriam Christian Mauduit) - PowerPoint PPT Presentation

WORKING WITH CHRISTIAN MAUDUIT (In memoriam Christian Mauduit) András Sárközy

2 1. THE BEGINNING AND THE CONTINUATION I met Christian first at a CIRM meeting. Then he invited me for a one month visit to the Institut de Mathématiques de Luminy in Marseille. This first visit of mine in 1994 was very fruitful and enjoyable. We ended up with a joint paper, and decided to continue the cooperation. Since then I have visited Marseille 24 times for 1, 2 or 3 month visits (for altogether 39 months) and Christian also visited me several times in Budapest. We have written 48 joint papers which make 26% of his papers and 18% of mine (in 31 of these papers we also had further coauthors). According to the citation statistics of the Mathematical Reviews the citation list of both of us is led by the same joint paper of ours, and the same 5 papers on the list (each of them written in pseudorandom binary sequences) out of the first 7 in my case and 6 in his case are our joint papers.

3 All our joint papers are written in number theory, and their subjects belong to one of the following 4 fields: digit properties, pseudorandom (to be abbreviated as PR) binary sequences, PR binary lattices, pseudorandomness in other situations. Now I will present a short survey of our papers grouping them according to their subjects and focusing on the most important ones.

4 2. DIGIT PROPERTIES Our first two papers with Christian (written in 1994 and 1995, resp.) were written on digit properties, more precisely, we studied the arithmetic structure of sets characterized by digit properties. This subject was proposed by Christian since he had been working earlier on problems of this flavour, but I was also interested in it since the digit properties can be controlled very well by using the generating function principle, and I was wondering: how well can one utilize this fact to study arithmetic properties? The first paper in this direction was written by Gelfond.

5 We continued Gelfond’s work by studying the distribution of the values S g ( a + b ) in residue classes where S g ( n ) denotes the sum of base g digits of n and a ∈ A , b ∈ B for two “dense” sets A , B of positive integers. We also studied the typical and extremal values of the function ω ( n ) ( = number of distinct prime factors of n ) over the integers n belonging to the set U r , m ( N ) = { n : n ≤ N , S g ( n ) ≡ r ( mod m ) } . In our second (much more difficult) paper we studied similar arithmetic properties of the integers n some of whose digits are fixed , i.e., which belong to the set V k ( N ) = { n : n ≤ N , S g ( n ) = k } (the difficulty is that this set is much thinner than the set U r , m ( N ) ).

6 In two triple papers with P. Erdős we studied similar arithmetic properties as Gelfond in his paper and two of us in our two earlier papers, but we considered even thinner sets characterized by digit properties: namely, we considered integers with “missing digits” , i.e., integers n such that in their base g representation a fixed digit does not occur. In a triple paper with Konyagin we also studied the number of prime factors of integers characterized by digit properties. In a triple paper with C. Dartyge we considered finite fields F q of order q = p r with r ≥ 2, and the analog of the base p representation of integers with “missing digits”. We showed that under certain conditions there are squares x 2 , more generally polynomial values f ( x ) , and generators g such that x 2 , f ( x ) and f ( g ) , resp., are of missing digits.

7 In two recent rather difficult and complicated triple papers with J. Rivat we returned to our early papers with Christian on the sum of digits function. In one of these new papers we proved a conjecture appearing in one of our early papers, while in the other new paper we sharpened some results proved in an other early paper of ours. 3. PSEUDORANDOM BINARY SEQUENCES We have written 17 papers on this subject whose study was proposed by me. As a student Szemerédi and I settled a combinatorial extremum problem of Erdős, and to show that the bound given for the extremal value can be achieved we used the Legendre symbol sequence and utilized (in a non-trivial way) its random-type structure.

8 I thought already then that this property can be also utilized elsewhere. Later I read about the Vernam cipher which is still a frequently applied encrypting algorithm and it is based on the use of PR binary sequences . I also heard a talk on H. Niederreiter’s work on PR sequences of real numbers from the interval [ 0 , 1 ) (to be used in numerical analysis). Based on these facts I suggested to Christian to try to develop a Niederreiter-type quantitative and constructive theory of PR binary sequences . He liked the idea, so that we started to work in this direction. First we did a lot of reading and made some inquiries, and it took us 3 months to end up with our fist paper of this type; this is probably our best paper and it is certainly the most frequently cited one.

9 In this paper we considered finite binary sequences of type E N = ( e 1 , e 2 , . . . , e N ) ∈ {− 1 , + 1 } N . (1) First we introduced the measures of pseudorandomness . The most important ones are the following: The well-distribution measure of the sequence E N in (1) is defined as t − 1 � � � � � W ( E N ) = max � , e a + jb � � a , b , t � j = 0 where the maximum is taken over all positive integers a , b , t such that 1 ≤ a ≤ a + ( t − 1 ) b ≤ N , while the correlation measure of order k of E N is defined as M � � � � � C k ( E N ) = max e n + d 1 e n + d 2 . . . e n + d k � , � � M , D � n = 1 where the maximum is taken over all D = ( d 1 , d 2 , . . . , d k ) and M such that d 1 , d 2 , . . . , d k and M are non-negative integers with 0 ≤ d 1 < d 2 < · · · < d k ≤ N − M .

10 We also defined the combined (well-distribution-correlation) PR-measure of order k : t � � � � � Q k ( E N ) = max e a + jb + d 1 e a + jb + d 2 . . . e a + jb + d k � � a , b , t , D � � j = 0 where a , b , t , ( 0 ≤ ) d 1 < d 2 < · · · < d k are positive integers, D = ( d 1 , d 2 , . . . , d k ) and we consider all the sums such that all the subscripts a + jb + d i belong to { 1 , 2 , . . . , N } . Then the sequence E N is considered as a “good” PR sequence if the measures W ( E N ) and C k ( E N ) (at least for “small” k ) are “small”. (This terminology was justified later.) We also proved in this paper (by using Weil’s theorem) that the Legendre symbol sequence �� 1 � � 2 � � p − 1 �� (2) E p − 1 = , , . . . , p p p is “good”. The paper described above was the first part of a series of 7 papers. In part II we applied the PR measures introduced in part I to study the PR properties of certain important special sequences:

11 We studied the Champernowne, Rudin–Shapiro and Thue–Morse sequences, and we also presented a further “good” PR sequence: we generalized the Legendre symbol sequence � � � � g ( n ) n construction (2) so that we replaced the n -th element by where g ( x ) is a permutation p p polynomial over F p satisfying a certain mild condition. Parts III and IV of the series were 5 author papers written jointly with J. Cassaigne, S. Ferenczi and J. Rivat on the PR properties of the Liouville function (defined as λ ( n ) = ( − 1 ) Ω( n ) where Ω( n ) denotes the number of prime factors of n counted with multiplicity). This is a difficult question whose importance is based on the fact that this function is closely related to the Möbius function playing a central role in multiplicative number theory.

12 In parts V and VI we considered a problem of Erdős. Let k be a positive integer, α an irrational number such that the partial quotients in its continued fraction expansion are bounded. Then we studied the PR properties of { n k α } , more precisely, of its distribution between the intervals [ 0 , 1 / 2 ) and [ 1 / 2 , 1 ) . In part VII of the series written jointly with J. Cassaigne we studied the measures of pseudorandomness defined earlier. We proved in this important paper that for a random binary sequence E N ⊂ {− 1 , + 1 } N with large probability we have N 1 / 2 ≪ W ( E N ) ≪ ( N log N ) 1 / 2 and N 1 / 2 ≪ C k ( E N ) ≪ ( kN log N ) 1 / 2 , so that, indeed, for a “good” PR sequence E N W ( E N ) and C k ( E N ) cannot be much larger than N 1 / 2 . We also studied the minimum values of the PR measures.

13 Another important part of this paper is the comparison of correlations of different order. We showed that if k | ℓ , N → ∞ and C ℓ ( E N ) is “small”, then C k ( E N ) also must be small, while if k ∤ ℓ , then C k ( E N ) and C ℓ ( E N ) are independent in the sense that either of them can be small, while the other one is large (so that the PR measures C 2 , C 3 , C 5 , . . . , C p , . . . are independent). In another paper with Christian we studied the connection between the measures W and C k . Jointly with J. Rivat we studied the PR properties of { n c } for n = 1 , 2 , . . . where c is not integer and c > 1, i.e., we considered its distribution between the intervals [ 0 , 1 / 2 ) , [ 1 / 2 , 1 ) for n = 1 , 2 , . . . , N . If one wants to apply PR binary sequences in cryptography then one has to construct large families of them (and possibly many constructions of this type). We have presented the following constructions: