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Upper and lower densities have the strong Darboux property
Full text linkLet P(N) be the power set of N. An upper density (on N) is a nondecreasing and subadditive function μ⋆:P(N)→R such that μ⋆(N)=1 and μ⋆(k⋅X+h)=1kμ⋆(X) for all X⊆N and h,k∈N+, where k⋅X+h:={kx+h:x∈X}. The upper asymptotic, upper Banach, upper logarithmic, upper Buck, upper Pólya, and upper analytic densities are some examples of upper densities. We show that every upper density μ⋆ has the strong Darboux property, and so does the associated lower density, where a function f:P(N)→R is said to have the strong Darboux property if, whenever X⊆Y⊆N and a∈[f(X),f(Y)], there is a set A such that X⊆A⊆Y and f(A)=a. In fact, we prove the above under the assumption that the monotonicity of μ⋆ is relaxed to the weaker condition that μ⋆(X)≤1 for every X⊆N
On small sets of integers
No full textAn upper quasi-density on H (the integers or the non-negative integers) is a real-valued subadditive function μ⋆ defined on the whole power set of H such that μ⋆(X) ≤ μ⋆(H) = 1 and μ⋆(k·X+h)=1kμ⋆(X) for all X⊆ H, k∈ N+, and h∈ N, where k· X: = { kx: x∈ X} ; and an upper density on H is an upper quasi-density on H that is non-decreasing with respect to inclusion. We say that a set X⊆ H is small if μ⋆(X) = 0 for every upper quasi-density μ⋆ on H. Main examples of upper densities are given by the upper analytic, upper Banach, upper Buck, and upper Pólya densities, along with the uncountable family of upper α-densities, where α is a real parameter ≥ - 1 (most notably, α= - 1 corresponds to the upper logarithmic density, and α= 0 to the upper asymptotic density). It turns out that a subset of H is small if and only if it belongs to the zero set of the upper Buck density on Z. This allows us to show that many interesting sets are small, including the integers with less than a fixed number of prime factors, counted with multiplicity; the numbers represented by a binary quadratic form with integer coefficients whose discriminant is not a perfect square; and the image of Z through a non-linear integral polynomial in one variable
On the number of distinct prime factors of a sum of super-powers
Full text linkGiven k,l∈N+, let xi,j be, for 1≤i≤k and 0≤j≤l some fixed integers, and define, for every n∈N+, sn:=∑i=1 k∏j=0 lxi,j n. We prove that the following are equivalent: (a) There are a real θ>1 and infinitely many indices n for which the number of distinct prime factors of sn is greater than the super-logarithm of n to base θ.(b) There do not exist non-zero integers a0,b0,...,al,bl such that s2n=∏i=0 lai (2n) and s2n−1=∏i=0 lbi (2n−1) for all n.We will give two different proofs of this result, one based on a theorem of Evertse (yielding, for a fixed finite set of primes S, an effective bound on the number of non-degenerate solutions of an S-unit equation in k variables over the rationals) and the other using only elementary methods. As a corollary, we find that, for fixed c1,x1,...,ck,xk∈N+, the number of distinct prime factors of c1x1 n+⋯+ckxk n is bounded, as n ranges over N+, if and only if x1=⋯=xk
On the density of sumsets
Full text linkRecently introduced by the authors in [Proc. Edinb. Math. Soc. 60 (2020),
139-167], quasi-densities form a large family of real-valued functions
partially defined on the power set of the integers that serve as a unifying
framework for the study of many known densities (including the asymptotic
density, the Banach density, the logarithmic density, the analytic density, and
the P\'olya density).
We further contribute to this line of research by proving that (i) for each
and , there is
with and for every
quasi-density and every , where is the
-fold sumset of and denotes the domain of definition
of ; (ii) for each and every non-empty finite
, there is with and for every quasi-density ; (iii)
for each , there exists with such that and for every
quasi-density .
Proofs rely on the properties of a little known density first considered by
R.C. Buck and the "structure" of the set of all quasi-densities; in particular,
they are rather different than previously known proofs of special cases of the
same results.Comment: 13 pages, to appear in Monatshefte f\"ur Mathematik. We fixed a gap
in the old "proof" of Theorem 3.1, which made it necessary to improve on
Proposition 2.4 (that is, Proposition 2.3 in the previous version of the
paper
On a system of equations with primes
Full text linkGiven an integer n ≥ 3, let u1, ..., un be pairwise coprime integers ≥ 2, D a family of nonempty proper subsets of {1, ..., n} with “enough” elements, and ε a function D → {±1}. Does there exist at least one prime q such that q divides ∏i∈I ui− ε(I) for some I ∈ D, but it does not divide u1 · · · un? We answer this question in the positive when the ui are prime powers and " and D are subjected to certain restrictions. We use the result to prove that, if ε0 ∈ {±1} and A is a set of three or more primes that contains all prime divisors of any number of the form ∏p∈B p−ε0 for which B is a finite nonempty proper subset of A, then A contains all the primes
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