16 research outputs found
Effective equidistribution and the Sato-Tate law for families of elliptic curves
Extending recent work of others, we provide effective bounds on the family of
all elliptic curves and one-parameter families of elliptic curves modulo p (for
p prime tending to infinity) obeying the Sato-Tate Law. We present two methods
of proof. Both use the framework of Murty-Sinha; the first involves only
knowledge of the moments of the Fourier coefficients of the L-functions and
combinatorics, and saves a logarithm, while the second requires a Sato-Tate
law. Our purpose is to illustrate how the caliber of the result depends on the
error terms of the inputs and what combinatorics must be done.Comment: Version 1.1, 24 pages: corrected the interpretation of Birch's moment
calculations, added to the literature review of previous results
The Norms of Graph Spanners
A -spanner of a graph is a subgraph in which all distances are
preserved up to a multiplicative factor. A classical result of Alth\"ofer
et al. is that for every integer and every graph , there is a
-spanner of with at most edges. But for some
settings the more interesting notion is not the number of edges, but the
degrees of the nodes. This spurred interest in and study of spanners with small
maximum degree. However, this is not necessarily a robust enough objective: we
would like spanners that not only have small maximum degree, but also have
"few" nodes of "large" degree. To interpolate between these two extremes, in
this paper we initiate the study of graph spanners with respect to the
-norm of their degree vector, thus simultaneously modeling the number
of edges (the -norm) and the maximum degree (the -norm).
We give precise upper bounds for all ranges of and stretch : we prove
that the greedy -spanner has norm of at most , and that this bound is tight (assuming the Erd\H{o}s girth
conjecture). We also study universal lower bounds, allowing us to give
"generic" guarantees on the approximation ratio of the greedy algorithm which
generalize and interpolate between the known approximations for the
and norm. Finally, we show that at least in some situations,
the norm behaves fundamentally differently from or
: there are regimes ( and stretch in particular) where
the greedy spanner has a provably superior approximation to the generic
guarantee
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
No distributed quantum advantage for approximate graph coloring
We give an almost complete characterization of the hardness of -coloring
-chromatic graphs with distributed algorithms, for a wide range of models
of distributed computing. In particular, we show that these problems do not
admit any distributed quantum advantage. To do that: 1) We give a new
distributed algorithm that finds a -coloring in -chromatic graphs in
rounds, with . 2) We prove that any distributed
algorithm for this problem requires rounds.
Our upper bound holds in the classical, deterministic LOCAL model, while the
near-matching lower bound holds in the non-signaling model. This model,
introduced by Arfaoui and Fraigniaud in 2014, captures all models of
distributed graph algorithms that obey physical causality; this includes not
only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL,
even with a pre-shared quantum state.
We also show that similar arguments can be used to prove that, e.g.,
3-coloring 2-dimensional grids or -coloring trees remain hard problems even
for the non-signaling model, and in particular do not admit any quantum
advantage. Our lower-bound arguments are purely graph-theoretic at heart; no
background on quantum information theory is needed to establish the proofs
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Visibility of lattice points in high dimensional spaces and extreme values of combinations of Dirichlet L-functions
This thesis is divided into three major topics. In the first, we study questions concerning the distribution of lattice points in dimensions two and higher. We give asymptotic formulas for the number of integer lattice points of fixed index visible from certain admissible sets. We also study the shape of the body when the lattice points move inside a given large ball, and is a lattice point visible from all 's.
In the second part, we study the phenomenon of similar ordering of Farey fractions and their generalizations. The notion of similar ordering for pairs of rationals was first introduced by Hardy, Littlewood and P\'olya. Later A.E. Mayer proved that pairs of Farey fractions in are similarly ordered when is large enough. We generalize Mayer's result to Ducci iterates of Farey sequences and visible points in convex regions. We also generalize the notion of index for Farey sequences and study the distribution of this generalization for large values of .
In the third part of this thesis, we work with large values of certain Dirichlet series and their partial sums. In particular, we examine a `short' partial sum of the Riemann Zeta function, . We obtain large values of for when with . We also adapt the methods by K. Soundararajan \cite{Sound} and A. Bondarenko, K. Seip \cite{Bond} to obtain large values on the critical line for a certain linear combination of Dirichlet -functions
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum