4 Citations
Multivariate Polynomial Values in Difference Sets
- Mathematics, Computer Science
- 2020
Every set lacking nonzero differences in $h(\mathbb{Z}^{\ell}) satisfies certain nonsingularity conditions, provided $h ($h) contains a multiple of every natural number and $h$ satisfies certain nontingingularity conditions.
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.
Quantitative bounds for Gowers uniformity of the M\"obius and von Mangoldt functions
- Mathematics
- 2021
We establish quantitative bounds on the U[N ] Gowers norms of the Möbius function μ and the von Mangoldt function Λ for all k, with error terms of shapeO((log logN)−c). As a consequence, we obtain…
References
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Difference sets and the primes
- Mathematics
- 2008
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 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…
Analytic Number Theory
- Mathematics
- 2004
Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large…
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…