Difference sets and the primes

  title={Difference sets and the primes},
  author={Imre Z. Ruzsa and Tom Sanders},
  journal={Acta Arithmetica},
Suppose that A is a subset of {1,...,N} 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(log N)^{1/4})) for some absolute c>0. 
Difference sets and the irreducibles in function fields
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On sets of polynomials whose difference set contains no squares
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A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes
Let k > 1 be a fixed integer, and PN be the set of primes no more than N . We prove that if set A ⊂ PN contains no patterns p1, p1 + (p2 − 1), where p1, p2 are prime numbers, then |A| |PN | ≪ ( log
On improving a Schur-type theorem in shifted primes
We show that if N ≥ exp(exp(exp(k))), then any k-colouring of the primes that are less than N contains a monochromatic solution to p1 − p2 = p3 − 1.
Difference sets and Polynomials of prime variables
Let \psi(x) be a polynomial with rational coefficients. Suppose that \psi has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper
Polynomials and Primes in Generalized Arithmetic Progressions
We provide upper bounds on the density of a symmetric generalized arithmetic progression lacking nonzero elements of the form h(n) for natural numbers n, or h(p) with p prime, for appropriate
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Problems and Results on Intersective Sets
By intersective set we mean a set H ⊂ Z having the property that it intersects the difference set A − A of any dense subset A of the integers. By analogy between the integers and the ring of
Positive exponential sums, difference sets and recurrence


Difference sets and shifted primes
We show that if A is a subset of {1, …, n} which has no pair of elements whose difference is equal to p − 1 with p a prime number, then the size of A is O(n(log log n)−c(log log log log log n)) for
On sets of natural numbers whose difference set contains no squares
We show that if a sequence s/ of natural numbers has no pair of elements whose difference is a positive square, then the density of J/ n{l,...,«} is O(l/log«) c »), cn->-oo. This improves previous
where the maximum is taken for those sets ul< u~-K.. , which form an &‘-set relative to the set 12, 22, . . . , n2, . . . . see [ll].) In the case of the arithmetic progressions of three terms, we
On measures of intersectivity
A set A of positive integers is called (difference) intersective, if A N ( B B ) # 0 whenever B has positive upper density. Here instead of upper density we might equally naturally write lower or
Integer Sets Containing No Arithmetic Progressions
lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of
Multiplicative Number Theory
From the contents: Primes in Arithmetic Progression.- Gauss' Sum.- Cyclotomy.- Primes in Arithmetic Progression: The General Modulus.- Primitive Characters.- Dirichlet's Class Number Formula.- The
Multiplicative Number Theory, 3rd ed., Grad
  • Texts in Math
  • 2000
Multiplicative number theory, volume 74 of Graduate Texts in Mathematics
  • Multiplicative number theory, volume 74 of Graduate Texts in Mathematics
  • 2000
Multiplicative number theory, volume 74 of Graduate Texts in Mathematics
  • Springer-Verlag, New York, third edition,
  • 2000