scholarly journals ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS

2018 ◽  
Vol 19 (4) ◽  
pp. 1259-1286
Author(s):  
Youness Lamzouri
Keyword(s):  
Algebraic Geometry ◽  
Lower Bound ◽  
Exponential Sums ◽  
Probabilistic Methods ◽  
Upper And Lower Bounds ◽  
Partial Sums ◽  
Log P ◽  
Analysis Techniques ◽  
Distribution Of The Maximum ◽  
Order Of Magnitude

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

SIMULATIONS OF UNARY ONE-WAY MULTI-HEAD FINITE AUTOMATA

2014 ◽  
Vol 25 (07) ◽  
pp. 877-896 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER ◽  
MATTHIAS WENDLANDT
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Finite Automaton ◽  
Finite Automata ◽  
Upper And Lower Bounds ◽  
Unary Languages ◽  
Order Of Magnitude ◽  
Simulation Results ◽  
Special Case ◽  
Nondeterministic Finite Automaton

We investigate the descriptional complexity of deterministic one-way multi-head finite automata accepting unary languages. It is known that in this case the languages accepted are regular. Thus, we study the increase of the number of states when an n-state k-head finite automaton is simulated by a classical (one-head) deterministic or nondeterministic finite automaton. In the former case upper and lower bounds that are tight in the order of magnitude are shown. For the latter case we obtain an upper bound of O(n2k) and a lower bound of Ω(nk) states. We investigate also the costs for the conversion of one-head nondeterministic finite automata to deterministic k-head finite automata, that is, we trade nondeterminism for heads. In addition, we study how the conversion costs vary in the special case of finite and, in particular, of singleton unary lanuages. Finally, as an application of the simulation results, we show that decidability problems for unary deterministic k-head finite automata such as emptiness or equivalence are LOGSPACE-complete.

The L1-norm of exponential sums in d

2013 ◽  
Vol 154 (3) ◽  
pp. 381-392 ◽  
Author(s):  
GIORGIS PETRIDIS
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Absolute Constant ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Correct Order ◽  
L1 Norm ◽  
Finite Set ◽  
Order Of Magnitude ◽  
Technical Sense

AbstractLet A be a finite set of integers and FA(x) = ∑a∈A exp(2πiax) be its exponential sum. McGehee, Pigno and Smith and Konyagin have independently proved that ∥FA∥1 ≥ c log|A| for some absolute constant c. The lower bound has the correct order of magnitude and was first conjectured by Littlewood. In this paper we present lower bounds on the L1-norm of exponential sums of sets in the d-dimensional grid d. We show that ∥FA∥1 is considerably larger than log|A| when A ⊂ d has multidimensional structure. We furthermore prove similar lower bounds for sets in , which in a technical sense are multidimensional and discuss their connection to an inverse result on the theorem of McGehee, Pigno and Smith and Konyagin.

scholarly journals Low Pseudomoments of the Riemann Zeta Function and Its Powers

2020 ◽  
Author(s):  
Maxim Gerspach
Keyword(s):  
Zeta Function ◽  
Lower Bounds ◽  
Riemann Zeta Function ◽  
Critical Line ◽  
Probabilistic Methods ◽  
Upper And Lower Bounds ◽  
Partial Sum ◽  
The Riemann Zeta Function ◽  
Order Of Magnitude ◽  
Riemann Zeta

Abstract The $2 q$-th pseudomoment $\Psi _{2q,\alpha }(x)$ of the $\alpha $-th power of the Riemann zeta function is defined to be the $2 q$-th moment of the partial sum up to $x$ of $\zeta ^\alpha $ on the critical line. Using probabilistic methods of Harper, we prove upper and lower bounds for these pseudomoments when $q \le \frac{1}{2}$ and $\alpha \ge 1$. Combined with results of Bondarenko et al., these bounds determine the size of all pseudomoments with $q> 0$ and $\alpha \ge 1$ up to powers of $\log \log x$, where $x$ is the length of the partial sum, and it turns out that there are three different ranges with different growth behaviours. In particular, the results give the order of magnitude of $\Psi _{2 q, 1}(x)$ for all $q> 0$.

scholarly journals A Lower Bound for the Query Phase of Contraction Hierarchies and Hub Labels and a Provably Optimal Instance-Based Schema

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 164
Author(s):  
Tobias Rupp ◽  
Stefan Funke
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Real World ◽  
Road Network ◽  
Upper And Lower Bounds ◽  
Bounded Degree ◽  
Shortest Path Routing ◽  
Graph Classes ◽  
Speed Up ◽  
To Come

We prove a Ω(n) lower bound on the query time for contraction hierarchies (CH) as well as hub labels, two popular speed-up techniques for shortest path routing. Our construction is based on a graph family not too far from subgraphs that occur in real-world road networks, in particular, it is planar and has a bounded degree. Additionally, we borrow ideas from our lower bound proof to come up with instance-based lower bounds for concrete road network instances of moderate size, reaching up to 96% of an upper bound given by a constructed CH. For a variant of our instance-based schema applied to some special graph classes, we can even show matching upper and lower bounds.

scholarly journals On Waring’s problem: Three cubes and a sixth power

2001 ◽  
Vol 163 ◽  
pp. 13-53 ◽  
Author(s):  
Jörg Brüdern ◽  
Trevor D. Wooley
Keyword(s):  
Exponential Sums ◽  
Large Integer ◽  
Natural Numbers ◽  
Waring’S Problem ◽  
Smooth Numbers ◽  
Waring's Problem ◽  
Order Of Magnitude ◽  
Almost All ◽  
Hardy Littlewood Method

We establish that almost all natural numbers not congruent to 5 modulo 9 are the sum of three cubes and a sixth power of natural numbers, and show, moreover, that the number of such representations is almost always of the expected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expected order of magnitude. Our results depend on a certain seventh moment of cubic Weyl sums restricted to minor arcs, the latest developments in the theory of exponential sums over smooth numbers, and recent technology for controlling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.

scholarly journals On the Rank of $p$-Schemes

10.37236/3097 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Fateme Raei Barandagh ◽  
Amir Rahnamai Barghi
Keyword(s):  
Lower Bound ◽  
Prime Number ◽  
Lower Bounds ◽  
Upper Bound ◽  
Upper And Lower Bounds ◽  
Minimum Rank

Let $n>1$ be an integer and $p$ be a prime number. Denote by $\mathfrak{C}_{p^n}$ the class of non-thin association $p$-schemes of degree $p^n$. A sharp upper and lower bounds on the rank of schemes in $\mathfrak{C}_{p^n}$ with a certain order of thin radical are obtained. Moreover, all schemes in this class whose rank are equal to the lower bound are characterized and some schemes in this class whose rank are equal to the upper bound are constructed. Finally, it is shown that the scheme with minimum rank in $\mathfrak{C}_{p^n}$ is unique up to isomorphism, and it is a fusion of any association $p$-schemes with degree $p^n$.

scholarly journals The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

2021 ◽  
Vol 157 (7) ◽  
pp. 1610-1651
Author(s):  
Pascal Autissier ◽  
Dante Bonolis ◽  
Youness Lamzouri
Keyword(s):  
Distribution Function ◽  
Recent Work ◽  
Character Sums ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Optimal Range ◽  
Uniform Estimates ◽  
The Third ◽  
Distribution Of The Maximum ◽  
Complex Valued

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$ -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

A Multiple Exponential Sum to Modulus p2

1985 ◽  
Vol 28 (4) ◽  
pp. 394-396 ◽  
Author(s):  
D. R. Heath-Brown
Keyword(s):  
Algebraic Geometry ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Total Degree ◽  
Image Position

AbstractFor suitable polynomials f(x) ∊ ℤ[x] in n variables, of total degree d, it is shown thatThis is, formally, a precise analogue of a theorem of Deligne [1] on exponential sums (mod p). However the proof uses no more than elementary algebraic geometry.

scholarly journals A Unifying View on Individual Bounds and Heuristic Inaccuracies in Bidirectional Search

2020 ◽  
Vol 34 (03) ◽  
pp. 2327-2334
Author(s):  
Vidal Alcázar ◽  
Pat Riddle ◽  
Mike Barley
Keyword(s):  
Lower Bound ◽  
Heuristic Search ◽  
Search Algorithm ◽  
Substantial Improvement ◽  
Full Potential ◽  
The Past ◽  
Heuristic Search Algorithms ◽  
Bidirectional Search ◽  
Order Of Magnitude ◽  
The Cost

In the past few years, new very successful bidirectional heuristic search algorithms have been proposed. Their key novelty is a lower bound on the cost of a solution that includes information from the g values in both directions. Kaindl and Kainz (1997) proposed measuring how inaccurate a heuristic is while expanding nodes in the opposite direction, and using this information to raise the f value of the evaluated nodes. However, this comes with a set of disadvantages and remains yet to be exploited to its full potential. Additionally, Sadhukhan (2013) presented BAE∗, a bidirectional best-first search algorithm based on the accumulated heuristic inaccuracy along a path. However, no complete comparison in regards to other bidirectional algorithms has yet been done, neither theoretical nor empirical. In this paper we define individual bounds within the lower-bound framework and show how both Kaindl and Kainz's and Sadhukhan's methods can be generalized thus creating new bounds. This overcomes previous shortcomings and allows newer algorithms to benefit from these techniques as well. Experimental results show a substantial improvement, up to an order of magnitude in the number of necessarily-expanded nodes compared to state-of-the-art near-optimal algorithms in common benchmarks.

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