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  • Author or Editor: András Sárközy x
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In a series of papers Mauduit and Sárközy studied finite pseudorandom sequences. In particular, in Part I they showed that the Legendre symbol forms an 'ideally good' pseudorandom sequence. In other parts of the series numerous other binary sequences have been tested for pseudorandomness but neither of them proved to be as 'good' as the Legendre symbol. Here another 'very good' sequence is constructed by using the notion of index.

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Summary By using the multiplicative inverse modulo p, a large family of finite binary sequences is constructed with strong pseudorandom properties. The crucial tool in the proofs is an (additive) character sum estimate of Eichenauer--Hermann and Niederreiter.

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Abstract  

The pseudorandom properties of finite binary sequences have been studied recently intensively. In the papers written on this subject the two distinct elements of the sequences are chosen equally with probability 1/2. In this paper the authors extend the work to the more general case when the two elements are chosen with probability p, resp. 1-p.

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Summary  

Recently, Goubin, Mauduit, Rivat and Srkzy have given three constructions for large families of binary sequences. In each of these constructions the sequence is defined by modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} congruences where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document} is a prime number. In this paper the three constructions are extended to the case when the modulus is of the form \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $p$ \end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $q$ \end{document} are two distinct primes not far apart (note that the well-known Blum-Blum-Shub and RSA constructions for pseudorandom sequences are also of this type). It is shown that these modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions also have certain strong pseudorandom properties but, e.g., the (``long range'') correlation of order \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $4$ \end{document} is large (similar phenomenon may occur in other modulo \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $pq$ \end{document} constructions as well).

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Authors: Cécile Dartyge, András Sárközy and Mihály Szalay
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Summary  

It is proved that the summands of almost all unequal partitions of nare well-distributed modulo dfor d=o(n 1/2).

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Authors: R. Ahlswede, L. Khachatrian and András Sárközy
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