Exercise 537 using manns theorem improve the result

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Exercise 5.3.7 . Using Mann’s Theorem, improve the result of Exercise 5.3.3 to the following: If A is a set of natural numbers containing 0 with δ ( A ) > 0, then A is a basis of order at most ceilingleft 1 ( A ) ceilingright . Exercise 5.3.8 (continuation). Show that for every 0 < α 1, there is a set A⊂ N with 0 ∈A and δ ( A ) = α which is a basis of order ceilingleft 1 ceilingright . Thus the estimate of the last exercise is best-possible. Suggestion: review Lepson’s construction. Exercise 5.3.9 . Let A = { a 1 < a 2 < ... } be an infinite collection of natural numbers with the following property: for every positive integer m , there exist infinitely many i for which ma i <a i +1 . Show that A is not an asymptotic basis of finite order. Exercise 5.3.10 . Using the result of the previous exercise, show that there exists a set A of natural numbers containing 0 and 1 whose counting function satisfies A ( N ) N 1 / 2 for all large N , but which is not a basis of finite order.
208 CHAPTER 5. A POTPOURRI OF ADDITIVE NUMBER THEORY Exercise 5.3.11 . Using Exercise 5.3.9, show that Theorem 5.3.9 is false if the condition that d ( A ) > 0 is replaced by d ( A ) > 0. Exercise 5.3.12 (Carroll [Car00]). Suppose the positive integers are partitioned into finitely many sets. Must one of them, say A i , have the property that A i ∪{ 0 } is an asymptotic basis of finite order? Exercise 5.3.13 (St¨ ohr [St¨ o55, Kriterium 8], ). Show that if a 1 ,a 2 ,... is a p -adically convergent sequence of integers, then { a i } is not an asymptotic basis of finite order. Prove the same for any finite union of p -adically convergent sequences (with respect to the same p ). Exercise 5.3.14 (Ostmann [Ost56, p.27]). For A⊂ N , let f ( A ) be the least integer h 2 for which h A∩Anegationslash = . That is, f ( A ) is the least h for which there exists a solution to a 1 + ··· + a h = a h +1 with each a i ∈A . Using Lemma 5.3.10, prove that f ( A ) exists for every infinite set A . Ostmann calls f ( A ) the Fermat index of A , since Fermat’s Last Theorem is equivalent to the assertion that f ( { n k : n = 1 , 2 ,... } ) > 2 for k 3. 5.4 Densities of Particular Sumsets Up to this point our investigation of sumsets and their densities has been rather general. We now turn to particular and particularly striking special cases. We commence our discussion with a result of Wirsing to the effect that if f is a “smooth” function whose second derivative satisfies certain (stringent) inequalities, then A := { f ( n ) : n = 1 , 2 ,... } satisfies d (2 A ) > 0. We then turn to Schnirelmann’s remarkable theorem that every n> 1 is a sum of at most C primes, for an absolute constant C . Central in the proof of Schnirelmann’s theorem is the proof that { p + q : p,q prime } has positive lower density. The next result we discuss, due to Ro- manov, has a similar flavor: for fixed a 2, the set { p + a k : p prime ,k 1 } has positive lower density. We present the proof of Romanov’s result as exposited by Nathanson [Nat96]. We then discuss some theorems of Erd˝os and Crocker related to the special case a = 2.

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