77 research outputs found
Memory Efficient Arithmetic
In this paper we give an algorithm for computing the mth base-b digit (m=1 is
the least significant digit) of an integer n (actually, it finds sharp
approximations to n/b^m mod 1), where n is defined as the last number in a
sequence of integers s1,s2,...,sL=n, where s1=0, s2=1, and each successive si
is either the sum, product, or difference of two previous sj's in the sequence.
In many cases, the algorithm will find this mth digit using far less memory
than it takes to write down all the base-b digits of n, while the number of bit
operations will grow only slighly worse than linear in the number of digits.
One consequence of this result is that the mth base-10 digit of 2^t can be
found using O(t^{2/3} log^C t) bits of storage (for some C>0), and O(t log^C t)
bit operations.
The algorithm is also highly parallelizable, and an M-fold reduction in
running time can be achieved using M processors, although the memory required
will then grow by a factor of M.Comment: Difference between this version and last: Better notation, light
corrections, and more explanation
A Combinatorial Method for Counting Smooth Numbers in Sets of Integers
In this paper we present a method for producing asymptotic estimates for the
number of integers in a given S having only ``small'' prime factors. The
conditions that need to be verified are simpler than those required by other
methods, and we apply our result to give an easy proof of a result which says
that dense subsets A and B of {1,2,...,x} always produce asymptotically the
expected number of x^r - smooth sums a+b, where a in A and b in B. Recall that
a number n is said to be y-smooth if all its prime divisors are at most y.Comment: Light Correction
An application of linear programming duality to discrete Fourier analysis and additive problems
Suppose that f is a function from Z_p -> [0,1] (Z_p is my notation for the
integers mod p, not the p-adics), and suppose that a_1,...,a_k are some places
in Z_p. In some additive number theory applications it would be nice to perturb
f slightly so that Fourier transform f^ vanishes at a_1,...,a_k, while additive
properties are left intact. In the present paper, we show that even if we are
unsuccessful in this, we can at least say something interesting by using the
principle of the separating hyperplane, a basic ingredient in linear
programming duality.Comment: This is a preliminary draft. Future drafts will have references,
cleaner proofs, and perhaps some applications of the main theore
On the Oscillations of Multiplicative Functions Taking Values +/- 1
For completely multiplicative functions f(n) taking values 1 and -1, under
certain conditions on f(n) we show that f(n) changes sign at least x exp(-7(log
log x)sqrt(log x)) times as n runs through the integers <= x.Comment: 12 pages, accepted by Journal of Number Theor
Arithmetic structures in smooth subsets of F_p
Fix integers a_1,...,a_d satisfying a_1 + ... + a_d = 0. Suppose that f : Z_N
-> [0,1], where N is prime. We show that if f is ``smooth enough'' then we can
bound from below the sum of f(x_1)...f(x_d) over all solutions (x_1,...,x_d) in
Z_N to a_1 x_1 + ... + a_d x_d == 0 (mod N). Note that d = 3 and a_1 = a_2 = 1
and a_3 = -2 is the case where x_1,x_2,x_3 are in arithmetic progression. By
``smooth enough'' we mean that the sum of squares of the lower order Fourier
coefficients of f is ``small'', a property shared by many naturally-occurring
functions, among them certain ones supported on sumsets and on certain types of
pseudoprimes. The paper can be thought of as a generalization of another result
of the author, which dealt with a F_p^n analogue of the problem. It appears
that the method in that paper, and to a more limited extent the present paper,
uses ideas similar to those of B. Green's ``arithmetic regularity lemma'', as
we explain in the paper.Comment: This is a very preliminary draft. Future drafts will have cleaner
proofs and tighter notatio
On the Structure of Sets with Few Three-Term Arithmetic Progressions
Fix a density d in (0,1], and let F_p^n be a finite field, where we think of
p fixed and n tending to infinity. Let S be any subset of F_p^n having the
minimal number of three-term progressions, subject to the constraint |S| is at
least dp^n. We show that S must have some structure, and that up to o(p^n)
elements, it is a union of a small number of cosets of a subspace of dimension
n-o(n).Comment: This is a much cleaner version of a proof published on the arxives
three years ago, but where this one holds for finite fields F_p^n. The result
in this paper is much clearer than that published previousl
The Minimal Number of Three-Term Arithmetic Progressions Modulo a Prime Converges to a Limit
Given a density t in (0,1], and a prime p, let S be any subset of F_p having
at least tp elements, and having the least number of three-term arithmetic
progressions mod p among all subsets of F_p with at least tp elements. Define
N(t,p) to be 1/p^2 times the number of three-term arithmetic progressions in S
modulo p. Note that N(t,p) does not depend on S -- it only depends on t and p.
An old result of Varnavides shows that for fixed t, N(t,p) > c(t) > 0 for all
primes p sufficiently large. But, does N(t,p) converge to a limit as p ->
infinity? We prove that it does.Comment: This draft is a significantly shorter proof of the theorem. To appear
in Canadian Math Bulleti
A Structure Theorem for Positive Density Sets Having the Minimal Number of 3-term Arithmetic Progressions
Assuming the well-known conjecture that [x,x+x^t] contains a prime for t > 0
and x sufficiently large, we prove: For 0 < r < 1, there exists 0 < s < r < 1,
0 < d < 1, and infinitely many primes q such that if S is a subset of Z/qZ
having density at least s, and having the least number of 3-term arithemtic
progressions among all sets of density at least s, then S is nearly translation
invariant in a very strong sense. Namely, there exists 0 <= b <= q-1 such that
|S intersect (S + bj)| = (1-g(s))|S|, for every 0 0 as
s -> 0. A curious feature of the proof is that Behrend's construction on large
subsets of {1,2,...,x} containing no 3-term a.p., is a key ingredient
On Non-intersecting Arithmetic Progressions
We prove that if one has k non-intersecting arithmetic progressions of
integers, with common differences 2 <= q_1,...,q_k <= x, then k < x exp((-1/6 +
o(1)) sqrt(log x loglog x)). This improves a result of Szemeredi and Erdos.Comment: Submitte
On the Decay of the Fourier Transform and Three Term Arithmetic Progressions
In this paper we prove a basic theorem which says that if f : F_p^n -> [0,1]
has the property that ||f^||_(1/3) is not too ``large''(actually, it also holds
for quasinorms 1/2-\delta in place of 1/3), and E(f) = p^{-n} sum_m f(m) is not
too ``small'', then there are lots of triples m,m+d,m+2d such that
f(m)f(m+d)f(m+2d) > 0. If f is the indicator function for some set S, then this
would be saying that the set has many three-term arithmetic progressions. In
principle this theorem can be applied to sets having very low density, where
|S| is around p^{n(1-c)} for some small c > 0.Comment: One small notational correction: In the paper I called ||f||_(1/3) a
`norm', when in fact it should be 'quasinorm'. This does not affect any
results, as I don't use the triangle inequality anywhere -- the 1/3 quasinorm
was only used as a convenient way to state a corollary of one of my result
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