Abstract. The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960 suggested looking at subsets without three-term geometric progressions, and constructed such a subset with density about 0.719. More recently, several authors have found upper bounds for the upper density of such sets. We significantly improve upon these bounds, and demonstrate a method of constructing sets with a greater upper density than Rankin's set. This construction is optimal in the sense that our method gives a way of effectively computing the greatest possible upper density of a geometric-progressionfree set. We also show that geometric progressions in Z/nZ behave more like Roth's theorem in that one cannot take any fixed positive proportion of the integers modulo a sufficiently large value of n while avoiding geometric progressions. BackgroundLet A be a subset of the positive integers. A three-term arithmetic progression in A is a progression (a, a + b, a + 2b) ∈ A 3 with b > 0, or equivalently, a solution to the equation a + c = 2b where a, b and c are distinct elements of A. We say that A is free of arithmetic progressions if it contains no such progressions. For any subset A of the positive integers we denote by d(A) its asymptotic density (if it exists) and its upper density byd(A).In 1952 Roth proved the following famous theorem [14]. Theorem 1.1 (Roth).If A is a subset of the positive integers withd(A) > 0 then A contains a 3-term arithmetic progression.In particular, Roth showed that for any fixed α > 0 and sufficiently large N, any subset of the integers {1, · · · , N} of size at least αN contains a 3-term arithmetic progression. We can also view Roth's result as a statement about arithmetic progressions (with 3 distinct elements) in the group of integers mod N. Namely, if we denote by D(Z/NZ) the size of the largest subset of Z/NZ free of arithmetic progressions, Roth's argument can be used to show Roth's theorem has since been generalized by Szemerédi to progressions of arbitrary length. Arithmetic-progression-free sets have also been studied in the context of arbitrary abelian groups. Meshulam [9] generalized Roth's theorem to finite abelian groups of odd order, and recently Lev [8] has extended Meshulam's ideas to arbitrary finite abelian groups, a result which will be needed later. Let D(G) be the size of the largest subset

We sharpen a 1980 theorem of Erdős and Wagstaff on the distribution of positive integers having a large shifted prime divisor. Specifically, we obtain precise estimates for the quantity N (x, y) := #{n ≤ x : − 1 | n for some − 1 > y, prime}, in essentially the full range of x and y. We then present an application to a problem in arithmetic statistics. Let T CM (d) denote the largest order of a torsion subgroup of a CM elliptic curve defined over a degree d number field. Recently, Bourdon, Clark, and Pollack showed that the set of d with T CM (d) > y has upper density tending to 0, as y → ∞. We quantify the rate of decay to 0, proving that the upper and lower densities of this set both have the form (log y) −β+o(1) , where β = 1 − 1+log log 2 log 2 (the Erdős-Ford-Tenenbaum constant).

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A set is primitive if no element of the set divides another. We consider primitive sets of monic polynomials over a finite field and find natural generalizations of many of the results known for primitive sets of integers. In particular we generalize a result of Besicovitch to show that there exist primitive sets in Fq[x] with upper density arbitrarily close to q−1 q . Then, for a primitive set A, we consider the sum a∈A 1 q deg a deg a , the natural analogue in this setting of a sum considered by Erdős for primitive subsets of the integers, and show that it is uniformly bounded over all primitive sets A. We end with a generalization of work of Martin and Pomerance on the asymptotic growth rate of the counting function of a primitive set. Along the way we prove a quantitative analogue of the Hardy-Ramanujan theorem for function fields, as well as bounds on the size of the k-th irreducible polynomial.

We consider the distribution of the largest prime divisor of the integers in the interval [2, x], and investigate in particular the mode of this distribution, the prime number(s) which show up most often in this list. In addition to giving an asymptotic formula for this mode as x tends to infinity, we look at the set of those prime numbers which, for some value of x, occur most frequently as the largest prime divisor of the integers in the interval [2, x]. We find that many prime numbers never have this property. We compare the set of "popular primes," those primes which are at some point the mode, to other interesting subsets of the prime numbers. Finally, we apply the techniques developed to a similar problem which arises in the analysis of factoring algorithms.