# Difference sets and the primes

@article{Ruzsa2008DifferenceSA, title={Difference sets and the primes}, author={Imre Z. Ruzsa and Tom Sanders}, journal={Acta Arithmetica}, year={2008}, volume={131}, pages={281-301} }

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.

## 29 Citations

Difference sets and the irreducibles in function fields

- Mathematics
- 2011

Let A be a subset of the polynomials of degree less than N over a finite field 픽q. Let r be any nonzero element of 픽q. We show that if the difference set A−A does not contain elements of the form…

On sets of polynomials whose difference set contains no squares

- Mathematics
- 2013

Let Fq[t] be the polynomial ring over the finite field Fq, and let GN be the subset of Fq[t] containing all polynomials of degree strictly less than N . Define D(N) to be the maximal cardinality of a…

A quantitative bound on Furstenberg-S\'ark\"ozy patterns with shifted prime power common differences in primes

- Mathematics
- 2021

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

- Mathematics
- 2021

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

- Mathematics
- 2007

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

- Mathematics
- 2013

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…

# A 85 INTEGERS 16 ( 2016 ) INTERSECTIONS OF SETS OF DISTANCE

- Mathematics
- 2016

We isolate conditions on the relative asymptotic size of sets of natural numbers A,B that guarantee a nonempty intersection of the corresponding sets of distances. Such conditions apply to a large…

Problems and Results on Intersective Sets

- Mathematics
- 2014

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

- MathematicsElectron. Notes Discret. Math.
- 2013

## References

SHOWING 1-10 OF 14 REFERENCES

Difference sets and shifted primes

- Mathematics, Computer Science
- 2007

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

- Mathematics
- 1988

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…

ON DIFFERENCE SETS OF SEQUENCES OF INTEGERS . III

- Mathematics
- 1978

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

- Mathematics
- 1984

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

- Mathematics
- 1987

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

- Mathematics
- 1967

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