Or Natural) Density D(A

Or Natural) Density D(A

FINE ASYMPTOTIC DENSITIES FOR SETS OF NATURAL NUMBERS MAURO DI NASSO Abstract. By allowing values in non-Archimedean extensions of the unit interval, we consider ¯nitely additive measures that generalize the asymptotic density. The existence of a natural class of such \¯ne densities" is independent of ZFC. Introduction The asymptotic (or natural) density d(A) for sets A of natural numbers is a central tool in number theory: jA \ [1; n]j d(A) = lim (provided the limit exists) n!1 n In many applications, it reveals useful to consider suitable extensions of d that are de¯ned for all subsets. Among the most relevant examples, there are the upper and lower density, the Schnirelmann density, and the upper Banach density (see e.g. [9], [7], [13]). Several authors investigated the general problem of densities, i.e. the possibil- ity of constructing ¯nitely additive measures that extend asymptotic density to all subsets of the natural numbers, and satisfy some additional properties (see e.g. [2], [11], [12] and [1]). Recently, generalized probabilities have been introduced that take values into non-Archimedean rings (see e.g. [8] and [10]). In this paper we pursue the idea of re¯ning the notion of density by allowing values into a non-Archimedean extension of the unit interval. To this aim, we introduce a notion of \¯ne density" as a suitable ¯nitely additive function on P(N) that gives a non-zero (in¯nitesimal) measure even to singletons. With the addition of a natural coherence property, such ¯ne densities are independent of ZFC: their existence is in fact equivalent to the existence of a special kind of P-point ultra¯lters on N. By simply taking a quotient, ¯ne densities yield non-atomic ¯nitely additive measures that { up to in¯nitesimals { agree with the asymptotic density, and that assign a non-zero measure to all and only the in¯nite sets. 2000 Mathematics Subject Classi¯cation. Primary 11B05; 03E05. Secondary 11R21. Key words and phrases. Asymptotic density, Ultra¯lter, Non-Archimedean group. 1 2 MAURO DI NASSO 1. Definition and first properties In this paper we follow a common practice in number theory, and denote by N the set of positive integers. Let R be an abelian linearly ordered group that extends the additive real line (R; +; <), and let [0; 1]R be its unit interval. De¯nition 1.1. A function d : P(N) ! [0; 1]R is a ¯ne asymptotic density, or simply a ¯ne density, if the following properties hold: (1) d(;) = 0 and d(N) = 1 ; (2) Finite Additivity: If A \ B = ;, then d(A [ B) = d(A) + d(B); (3) Monotonicity: If jA \ [1; n]j · jB \ [1; n]j for all n, then d(A) · d(B); (4) Subset Property: If d(A) · d(B) then d(A) = d(B0) for a suitable subset B0 ⊆ B ; (5) Fineness: d(fng) = d(fmg) = " > 0 for all n; m 2 N. Clearly, the common density " of all singletons is the smallest possible non- zero density, i.e. d(A) < " if and only if A = ;. Notice also that whenever A ½ B is a proper inclusion, d(A) < d(B). If A = fa1; : : : ; ang ½ N is a ¯nite set of cardinality n, then Xn 1 = d(N) < d(A) = d(faig) = n ¢ ": i=1 In particular, the number " 2 [0; 1]R (as well as the ¯ne density of any ¯nite set) is in¯nitesimal. So, R is a non-Archimedean group, i.e. it contains positive numbers " > 0 such that " < 1=n for all n 2 N. We say that two elements »; ´ 2 [0; 1]R are in¯nitely close, and write » ¼ ´, if j» ¡ ´j is in¯nitesimal. Since R extends R, every » 2 [0; 1]R is in¯nitely close to a unique real number r 2 [0; 1] (just take r = inffx 2 [0; 1] j » · xg). We call such a number r ¼ » the standard part of », and write r = st(»). Note that st(» + ³) = st(») + st(³). As a ¯rst result, we show that { up to in¯nitesimals { the congruence classes have the expected ¯ne densities. m m Proposition 1.2. Let Ci = fmn + i j n 2 Ng. Then st(d(Ci )) = 1=m for all i = 1; : : : ; m. Proof. The following inequalities hold for all n: m m m m jCm \ [1; n]j · jCm¡1 \ [1; n]j · ::: · jC1 \ [1; n]j · j(Cm [ f1g) \ [1; n]j: So, by monotonicity, m m m m m m d(Cm ) · d(Cm¡1) · ::: · d(C1 ) · d(Cm [ f1g) = d(Cm ) + d(f1g) ¼ d(Cm ): FINE ASYMPTOTIC DENSITIES 3 m m If r = st(d(Cm )), then st(d(Ci )) = r for all i = 1; : : : ; m. By additivity: Ã ! Xm Xm m m 1 = st(d(N)) = st d(Ci ) = st(d(Ci )) = m ¢ r; i=1 i=1 and the proof is complete. ¤ We are now ready to prove that ¯ne densities actually generalize the asymp- totic density. For A ⊆ N and n 2 N, denote by An = jA \ [1; n]j the number of elements in A that are not greater than n. Proposition 1.3. 0 0 (1) If r · An=n · r for all n ¸ k, then r · st(d(A)) · r . (2) st(d(A)) is a limit point of the sequence hAn=n j n 2 Ni. (3) st(d(A)) = d(A) whenever A has asymptotic density d(A). Proof. (1). First of all, notice that we can assume without loss of generality that 0 k = 1, i.e. that the inequalities r · An=n and An=n · r hold for all n 2 N. In fact, let h = Ak, and set: A+ = (A n [1; k]) [ [1; h] and A¡ = (A n [1; k]) [ [k ¡ h + 1; k]: Then d(A \ [1; k]) = d([1; h]) = d([k ¡ h + 1; k]) and so d(A) = d(A+) = d(A¡). Moreover, for all n: A A+ A¡ A r · n · n and n · n · r0: n n n n Now let the rational numbers 0 · p=q · r and r0 · p0=q0 · 1 be ¯xed, and consider the subsets [q [p0 q q0 X = Ci and Y = Ci : i=q¡p+1 i=1 0 Notice that Xn · nr · An · nr · Yn for all n. By the properties of a ¯ne density, and by the previous proposition, we obtain 0 q p 0 p X X 0 p ¼ d(Cq) = d(X) · d(A) · d(Y ) = d(Cq ) ¼ : q i i q0 i=q¡p+1 i=1 As this is true for all fractions 0 · p=q · r and all fractions r0 · p0=q0 · 1, it follows that r · st(d(A)) · r0. ¡ + (2). Denote by simplicity an = An=n, and let l = lim infn!1 an and l = ¡ + ¡ lim supn!1 an. The sets fn j an < l g and fn j l < ang are ¯nite, and so l · + d(A) · l by (1). Now notice that jan+1 ¡ anj < 1=n for all n. As a consequence, ¡ + any real number in the interval [l ; l ] is a limit point of the sequence han j n 2 Ni. In particular, this applies to st(d(A)). 4 MAURO DI NASSO (3). It directly follows from (2). ¤ 2. The underlying ultrafilter Throughout Section 1, we never used the ² Subset Property: If d(A) · d(B) then d(A) = d(B0) for a suitable subset B0 ⊆ B. We remark that this natural assumption is needed to prove useful simple facts, such as the implication: d(A) < d(B) ) d(A)+" · d(B). Most notably, as shown below, the subset property allows for a proof that every ¯ne density carries a non-principal ultra¯lter. For X ⊆ N, we adopt the following notation: ² Xc = N n X is the complement of X. ² X + 1 = fx + 1 j x 2 Xg is the unit right-translation of X. Notice that for every n,(X + 1)n · Xn · ((X + 1) [ f1g)n. By monotonicity, d(X + 1) · d(X) · d(X + 1) + ", and so either d(X) = d(X + 1) or d(X) = d(X + 1) + ". Proposition 2.1. The family U d = f X ⊆ N j d(X) = d(X + 1) + " g is a non-principal ultra¯lter on N. Proof. We ¯rst prove the following: Claim. In every partition N = X [ Y [ Z, exactly one of the three pieces belongs to U d. Since N = f1g [ (X + 1) [ (Y + 1) [ (Z + 1) is a partition, by additivity: d(X) + d(Y ) + d(Z) = " + d(X + 1) + d(Y + 1) + d(Z + 1); and the claim follows. c As ; = ; + 1, we have that ; 2= U d. By taking X = A, Y = A and Z = ;, c the above claim implies that A2 = U d , A 2 U d. Now let A; B 2 U d, and c consider the partition X = A, Y = B n A, and Z = (A [ B) . Since A 2 U d, c it must be B n A2 = U d. The sets B n A, B and A \ B form a partition where c B n A2 = U d and B 2= U d; then A \ B 2 U d. If B ¶ A 2 U d, then also B 2 U d, c c otherwise B 2 U d would imply A \ B = ; 2 U d, a contradiction. Finally, U d is non-principal because for every n, d(fng) = d(fn + 1g) = d(fng + 1). ¤ In order to prove the next result, we need an additional natural property. De¯nition 2.2.

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    11 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us