801 research outputs found
The Frobenius problem, rational polytopes, and Fourier-Dedekind Sums
We study the number of lattice points in integer dilates of the rational
polytope ,
where are positive integers. This polytope is closely related to
the linear Diophantine problem of Frobenius: given relatively prime positive
integers , find the largest value of t (the Frobenius number) such
that has no solution in positive integers
. This is equivalent to the problem of finding the largest dilate
tP such that the facet contains no lattice point. We
present two methods for computing the Ehrhart quasipolynomials of P which count
the integer points in the dilated polytope and its interior. Within the
computations a Dedekind-like finite Fourier sum appears. We obtain a
reciprocity law for these sums, generalizing a theorem of Gessel. As a
corollary of our formulas, we rederive the reciprocity law for Zagier's
higher-dimensional Dedekind sums. Finally, we find bounds for the
Fourier-Dedekind sums and use them to give new bounds for the Frobenius number.Comment: Added journal referenc
Refinements of G\'al's theorem and applications
We give a simple proof of a well-known theorem of G\'al and of the recent
related results of Aistleitner, Berkes and Seip [1] regarding the size of GCD
sums. In fact, our method obtains the asymptotically sharp constant in G\'al's
theorem, which is new. Our approach also gives a transparent explanation of the
relationship between the maximal size of the Riemann zeta function on vertical
lines and bounds on GCD sums; a point which was previously unclear. Furthermore
we obtain sharp bounds on the spectral norm of GCD matrices which settles a
question raised in [2]. We use bounds for the spectral norm to show that series
formed out of dilates of periodic functions of bounded variation converge
almost everywhere if the coefficients of the series are in , with . This was previously known with ,
and is known to fail for . We also develop a sharp Carleson-Hunt-type
theorem for functions of bounded variations which settles another question
raised in [1]. Finally we obtain almost sure bounds for partial sums of dilates
of periodic functions of bounded variations improving [1]. This implies almost
sure bounds for the discrepancy of with an arbitrary growing
sequences of integers.Comment: 16 page
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