Difference sets and shifted primes
نویسندگان
چکیده
منابع مشابه
Difference Sets and the Primes
Suppose that A ⊂ {1, . . . , N} is such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(−c 4 √ logN)) for some absolute c > 0.
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Abstract. We bound from below the number of shifted primes p+s ≤ x that have a divisor in a given interval (y, z]. Kevin Ford has obtained upper bounds of the expected order of magnitude on this quantity as well as lower bounds in a special case of the parameters y and z. We supply here the corresponding lower bounds in a broad range of the parameters y and z. As expected, these bounds depend h...
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Best possible bounds are obtained for the concentration function of an additive arithmetic function on sequences of shifted primes. A real-valued function / defined on the positive integers is additive if it satisfies f(rs) = f(r) + f(s) whenever r and s are coprime. Such functions are determined by their values on the prime-powers. For additive arithmetic function /, let Q denote the frequency...
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We consider shifts of a set A ⊆ N by elements from another set B ⊆ N, and prove intersection properties according to the relative asymptotic size of A and B. A consequence of our main theorem is the following: If A = {an} is such that an = o(n k/k−1), then the k-recurrence set Rk(A) = {x | |A∩ (A+ x)| > k} contains the distance sets of some arbitrarily large finite sets.
متن کاملMultiple Recurrence and Convergence for Certain Averages along Shifted Primes
We show that any subset A ⊂ N with positive upper Banach density contains the pattern {m,m + [nα], . . . ,m + k[nα]}, for some m ∈ N and n = p − 1 for some prime p, where α ∈ R\Q. Making use for the Furstenberg Correspondence Principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averag...
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ژورنال
عنوان ژورنال: Acta Mathematica Hungarica
سال: 2008
ISSN: 0236-5294,1588-2632
DOI: 10.1007/s10474-007-7107-1