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

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

Subsets of F_p^n without three term arithmetic progressions have several large Fourier coefficients

Suppose that f : F_p^n -> [0,1] has expected value t in [p^(-n/9),1] (so, the density t can be quite low!). Furthermore, suppose that support(f) has no three-term arithmetic progressions. Then, we develop non-trivial lower bounds for f_j, which is the jth largest Fourier coefficient of f. This result is similar in spirit to that appearing in an earlier paper [1] by the author; however, in that paper the focus was on the ``small'' Fourier coefficients, whereas here the focus is on the ``large'' Fourier coefficients. Furthermore, the proof in the present paper requires much more sophisticated arguments than those of that other paper.Comment: This is a preliminary draft. Later drafts will have more references and cleaner proof

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

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

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

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

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