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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.
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