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For r-~2 let p(n, r) denote the maximum cardinality of a subset A of N={1, 2 .... , n} such that there are no Bc A and an integer y with S b=y'. It is shown that for any e >-0 and bEB n>-n(e), (l+o(l))2~/t'+l>n('-l>/t'+l)~_p(n, r)~_n~ § for all r_~5, and that for every fixed r~_6, p(n,r)=(l+o(1)).21/t'+~)n (~-1)/(' § as n~. Let f(n,m) denote… (More)

We develop a unified approach to the problem of clustering in the three different fields of applications indicated in the title of the paper, in the case when the parametric function of the models is regularly varying with positive exponent. The approach is based on Khintchine's probabilistic method that grew out of the Darwin-Fowler method in statistical… (More)

The spectral-null code S(n; k) of k-th order and length n is the union of n-tuples with 1 components, having k-th order spectral null at zero frequency. We determine the exact asymptotic in n behaviour of the size of such codes. In particular, we prove that for n satisfying some divisibility conditions, log 2 jS(n; k)j = n ? k 2 2 log 2 n + c k + o(1),… (More)

We study the connection between the additive structure of a finite set A ⊂ Z and the measure of large values of a trigonometric sum.

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