• Corpus ID: 11969516


  author={A. Saricozy},
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 may use the following simple fact: (i) A set afqrr,, afqu,, “,. , afqu, (where a is an integer and t, q, ul, u2, . . . , u, are positive integers) does not contain an arithmetic progression of three terms if and only if also the set u,, u,, . . . , U, has this property. 
On arithmetic structures in dense sets of integers
We prove that if A⊆ {1, . . . , N} has density at least (log logN)−c, where c is an absolute constant, then A contains a triple (a,a+d,a+2d) with d = x2+ y2 for some integers x, y, not both zero. We
On Infinite-Difference Sets
1. Introduction. Let A be a sequence; throughout this paper sequences are understood to be infinite, strictly increasing and composed of non-negative integers. We define D, the infinite-difference
Lower bounds in the polynomial Szemer\'edi theorem.
We construct large subsets of the first $N$ positive integers which avoid certain arithmetic configurations. In particular, we construct a set of order $N^{0.7685}$ lacking the configuration
The primes contain arbitrarily long polynomial progressions
We establish the existence of infinitely many polynomial progressions in the primes; more precisely, given any integer-valued polynomials P1, …, Pk ∈ Z[m] in one unknown m with P1(0) = … = Pk(0) = 0,
Difference sets and the irreducibles in function fields
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
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
Multiple recurrence and convergence for sequences related to the prime numbers
For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the
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
Interpolation sets and nilsequences
  • A. Le
  • Mathematics
    Colloquium Mathematicum
  • 2020
To give positive answer to a question of Frantzikinakis, we study a class of subsets of $\mathbb{N}$, called interpolation sets, on which every bounded sequence can be extended to an almost periodic


The exceptional set in Goldbach''s problem
An electromagnetic release device for cameras which may selectively be operated continuously comprises a motor circuit, a first switching means adapted to turn the motor circuit on or off, a stopper
Sur quelques ensembles d’entiers
  • fomptes Rendus, 234
  • 1952
ROTH, lrregularities of sequences relative to arithmetic progressions, I-IV
  • Marh. Ann,,
  • 1967
ROT~, On certain sets of integers,
  • London Math. Soc.,
  • 1953
VAUGHAN, The exceptional set in Goldbach’s problem, Acta Arithmetica
  • 1975
ROTH, Iil:egularities of sequences relative to arithmetic progressions
  • I--IV, _Math. Ann.,
  • 1967