Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-27T07:34:45.644Z Has data issue: false hasContentIssue false
  • English
  • Français

A Marstrand Theorem for Subsets of Integers

Published online by Cambridge University Press:  25 October 2013

YURI LIMA  and
CARLOS GUSTAVO MOREIRA
YURI LIMA
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA (e-mail: [email protected])
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brasil (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose a counting dimension for subsets of $\mathbb{Z}$ and prove that, under certain conditions on E,F ⊂ $\mathbb{Z}$, for Lebesgue almost every λ ∈ $\mathbb{R}$ the counting dimension of E + ⌊λF⌋ is at least the minimum between 1 and the sum of the counting dimensions of E and F. Furthermore, if the sum of the counting dimensions of E and F is larger than 1, then E + ⌊λF⌋ has positive upper Banach density for Lebesgue almost every λ ∈ $\mathbb{R}$. The result has direct consequences when E,F are arithmetic sets, e.g., the integer values of a polynomial with integer coefficients.

Keywords

Primary 05A99Secondary 37E15
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 23 , Issue 1 , January 2014 , pp. 116 - 134
Copyright
Copyright © Cambridge University Press 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bedford, T. and Fisher, A. (1992) Analogues of the Lebesgue density theorem for fractal sets of reals and integers. Proc. London Math. Soc. (3) 64 95–124.Google Scholar
[2]Bergelson, V. and Leibman, A. (1996) Polynomial extensions of van der Waerden's and Szemerédi's theorems. J. Amer. Math. Soc. 9 725–753.Google Scholar
[3]Bourgain, J. (1989) Pointwise ergodic theorems for arithmetic sets. Publ. Math. IHES 69 5–45.Google Scholar
[4]Fisher, A. (1993) Integer Cantor sets and an order-two ergodic theorem. Ergodic Theory and Dynamical Systems 13 45–64.Google Scholar
[5]Furstenberg, H. (1970) Intersections of Cantor sets and transversality of semigroups. In Problems in Analysis: Sympos. Salomon Bochner, Princeton University, 1969, pp. 41–59.Google Scholar
[6]Green, B. and Tao, T. (2008) The primes contain arbitrarily long arithmetic progressions. Ann. of Math. 167 481–547.Google Scholar
[7]Kitchens, B. (1997) Symbolic Dynamics: One-Sided, Two-Sided and Countable State Markov Shifts, Springer.Google Scholar
[8]Lima, Y. and Moreira, C. G. (2011) A combinatorial proof of Marstrand's theorem for products of regular Cantor sets. Expositiones Mathematicae 29 (2)231–239.Google Scholar
[9]Lima, Y. and Moreira, C. G. (2011) Yet another proof of Marstrand's theorem. Bull. Brazil. Math. Soc. 42 (2)331–345.Google Scholar
[10]Marstrand, J. M. (1954) Some fundamental geometrical properties of plane sets of fractional dimensions. Proc. London Math. Soc. 3 257–302.Google Scholar
[11]Mattila, P. (1995) Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press.Google Scholar
[12]Naudts, J. (1988) Dimension of discrete fractal spaces. J. Phys. A 21 447–452.Google Scholar
[13]Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 199–245.Google Scholar