Article

# Exponential Sums and Distinct Points on Arcs

01/2009;
Source: arXiv

ABSTRACT Suppose that some harmonic analysis arguments have been invoked to show that the indicator function of a set of residue classes modulo some integer has a large Fourier coefficient. To get information about the structure of the set of residue classes, we then need a certain type of complementary result. A solution to this problem was given by Gregory Freiman in 1961, when he proved a lemma which relates the value of an exponential sum with the distribution of summands in semi-circles of the unit circle in the complex plane. Since then, Freiman's Lemma has been extended by several authors. Rather than residue classes, one has considered the situation for finitely many arbitrary points on the unit circle. So far, Lev is the only author who has taken into consideration that the summands may be bounded away from each other, as is the case with residue classes. In this paper we extend Lev's result by lifting a recent result of ours to the case of the points being bounded away from each other.

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• ##### Article:Large sum-free sets in ℤ/pℤ
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ABSTRACT: We show that ifp is prime andA is a sum-free subset of ℤ/ p ℤ withn:=|A|>0.33p, thenA is contained in a dilation of the interval [n,p−n] (modp).
Israel Journal of Mathematics 04/2012; 154(1):221-233. · 0.75 Impact Factor
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##### Article:Distribution of points on the circle
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ABSTRACT: In connection with the proof of his celebrated “2.4-Theorem”, Freiman proved that if α1,…,αN are real numbers such that each interval [u,u+1/2) contains at most n of the αj mod 1, then . Freiman's result was extended by Moran and Pollington, and recently by Lev. This paper contains further extensions.
Journal of Number Theory.
• Inverse problems of additive number theory On the addition of sets of residues with respect to a prime modulus (Russian). G A Freiman . 1961. Doklady Akad. Nauk SSSR 141 571-573.

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25 Oct 2012

### Keywords

arbitrary points

certain type

complementary result

exponential sum

Freiman's Lemma

Gregory Freiman

harmonic analysis arguments

large Fourier coefficient

lemma

Lev

Lev's result

recent result

residue classes

residue classes modulo

semi-circles

summands

unit circle