3,962 research outputs found
Sets with small sumset and rectification
We study the extent to which sets A in Z/NZ, N prime, resemble sets of
integers from the additive point of view (``up to Freiman isomorphism''). We
give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and
|A| < c(K)N then A is Freiman isomorphic to a set of integers. Because we avoid
appealing to Freiman's structure theorem, we get a reasonable bound: we can
take c(K) > exp(-cK^2 log K).
As a byproduct of our argument we obtain a sharpening of the second author's
result on sets with small sumset in torsion groups. For example if A is a
subset of F_2^n, and if |A + A| < K|A|, then A is contained in a coset of a
subspace of size no more than 2^{CK^2}|A|.Comment: 9 pages, minor corrections mad
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
Terahertz generation in Czochralski grown periodically poled Mg:Y:LiNbO3 via optical rectification
Using a canonical pump-probe experimental technique, we studied the terahertz
(THz) waves generation and detection via optical rectification and mixing in
Czochralski-grown periodically poled Mg:Y:LiNbO3 (PPLN) crystals. THz waves
with frequencies at 1.37 THz and 0.68 THz as well as 1.8 THz were obtained for
PPLN with nonlinear grating periods of 0.03 and 0.06 mm, respectively. A
general theoretical model was developed by considering the dispersion and
damping of low frequency phonon-polariton mode. Our results show that THz waves
are generated in forward and backward directions via pumping pulse
rectification. The generated THz waves depend on the spectral shape of the
laser pulses, quasi-phase mismatches and dispersion characteristics of a
crystal.Comment: 25 pages, 4 figure
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