2,894 research outputs found
On sets of integers which contain no three terms in geometric progression
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-progression-free
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.Comment: 16 page
What is good mathematics?
Some personal thoughts and opinions on what ``good quality mathematics'' is,
and whether one should try to define this term rigorously. As a case study, the
story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So
On the maximal number of three-term arithmetic progressions in subsets of Z/pZ
Let a be a real number between 0 and 1. Ernie Croot showed that the quantity
\max_A #(3-term arithmetic progressions in A)/p^2, where A ranges over all
subsets of Z/pZ of size at most a*p, tends to a limit as p tends to infinity
through primes. Writing c(a) for this limit, we show that c(a) = a^2/2 provided
that a is smaller than some absolute constant. In fact we prove rather more,
establishing a structure theorem for sets having the maximal number of 3-term
progressions amongst all subsets of Z/pZ of cardinality m, provided that m <
c*p.Comment: 12 page
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