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61
Explicit Bounds on Exponential Sums and the Scarcity of Squarefree Binomial Coefficients
, 1996
"... This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involv ..."
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Cited by 15 (1 self)
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This paper fills what we believe to be a lacuna in the existing literature concerning upper bounds on exponential sums. Although it has always been evident that many of the known estimates can be made explicit, it is a nontrivial problem to actually do so. In particular so that the constants involved do not render the explicit estimates useless in practical applications. We have used the practical bounds that are needed to prove Theorem 1 as motivation for our results here, though we hope that this work will be applicable to a variety of other problems which routinely apply these or related exponential sum estimates. In particular our results here can be used to say something about the questions of estimating the number of integers free of large prime factors in short intervals (see [FL]), and of the largest prime factor of an integer in an interval (see [J]). Our key result is
Monochromatic equilateral right triangles on the integer grid
"... For any coloring of the N × N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d). ..."
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Cited by 11 (0 self)
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For any coloring of the N × N grid using less than log log n colors, one can always find a monochromatic equilateral right triangle, a triangle with vertex coordinates (x, y), (x + d, y), and (x, y + d).
SIEVING BY LARGE INTEGERS AND COVERING SYSTEMS OF CONGRUENCES
, 2006
"... An old question of Erdős asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if � n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirm ..."
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Cited by 11 (0 self)
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An old question of Erdős asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) (mod n) for n ∈ S whose union is Z. We prove that if � n∈S 1/n is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdős and Selfridge. We also prove a conjecture of Erdős and Graham, that, for each fixed number K> 1, the complement in Z of any union of residue classes r(n) (mod n), for distinct n ∈ (N, KN], has density at least dK for N sufficiently large. Here dK is a positive number depending only on K. Either of these new results implies another conjecture of Erdős and Graham, that if S is a finite set of moduli greater than N, with a choice for residue classes r(n) (mod n) for n ∈ S which covers Z, then the largest member of S cannot be O(N). We further obtain stronger forms of these results and establish other information, including an improvement of a related theorem of Haight. 1
Long arithmetic progressions in sumsets: thresholds and bounds
"... One of the main tasks of additive number theory is to examine structural properties of sumsets. For a set A of integers, the sumset lA = A + ···+ A consists of those numbers which can be represented as a sum of l elements of A: lA = {a1 + ···+ alai ∈ Ai}. Closely related and equally interesting not ..."
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Cited by 9 (4 self)
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One of the main tasks of additive number theory is to examine structural properties of sumsets. For a set A of integers, the sumset lA = A + ···+ A consists of those numbers which can be represented as a sum of l elements of A: lA = {a1 + ···+ alai ∈ Ai}. Closely related and equally interesting notion is that of l∗A,whichisthecol lection of numbers which can be represented as a sum of l different elements of A: l ∗ A = {a1 + ···+ alai ∈ Ai,ai � = aj}. Among the most wellknown results in all of mathematics are Vinogradov’s theorem, which says that 3P (P is the set of primes) contains all sufficiently large odd numbers, and Waring’s conjecture (proved by Hilbert, Hardy and Littlewood, Hua, and many others), which asserts that for any given r, there is a number l such that l∗Nr (Nr denotes the set of rth powers) contains all sufficiently large positive integers (see [29] for an excellent exposition concerning these results). In recent years, a considerable amount of attention has been paid to the study of finite sumsets. Given a finite set A and a positive integer l, the natural analogue
Finite and Infinite Arithmetic Progressions in Sumsets
, 2003
"... We prove that if A is a subset of at least cn 1/2 elements of {1,...,n}, where c is a sufficiently large constant, then the collection of subset sums of A contains an arithmetic progression of length n. As an application, we confirm a long standing conjecture of Erdős and Folkman on complete sequenc ..."
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Cited by 8 (3 self)
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We prove that if A is a subset of at least cn 1/2 elements of {1,...,n}, where c is a sufficiently large constant, then the collection of subset sums of A contains an arithmetic progression of length n. As an application, we confirm a long standing conjecture of Erdős and Folkman on complete sequences.
Noncanonical extensions of ErdősGinzburgZiv theorem
 Integers
, 2002
"... In 1961, ErdősGinzburgZiv proved that for a given natural number n ≥ 1 and a sequence a1,a2, ···,a2n−1 of integers (not necessarily distinct), there exist 1 ≤ i1 <i2 < ·· · < in ≤ 2n −1 such that ai1 +ai2 + ···+ain is divisible by n. Moreover, the constant 2n −1 is tight. By now, there ar ..."
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Cited by 7 (1 self)
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In 1961, ErdősGinzburgZiv proved that for a given natural number n ≥ 1 and a sequence a1,a2, ···,a2n−1 of integers (not necessarily distinct), there exist 1 ≤ i1 <i2 < ·· · < in ≤ 2n −1 such that ai1 +ai2 + ···+ain is divisible by n. Moreover, the constant 2n −1 is tight. By now, there are many canonical generalizations of this theorem. In this paper, we shall prove some noncanonical generalizations of this theorem.
A discrete Fourier kernel and Fraenkel’s tiling conjecture, Acta Arith
, 2005
"... The set B q p,r: = {⌊nq/p + r ⌋ : n ∈ Z} (with integers p,q,r) is a Beatty set with density p/q. We derive a formula for the Fourier transform ̂B q p∑ p,r(j): = e −2πij⌊nq/p+r⌋/q. n=1 A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m ≥ 3 Beatty sets with d ..."
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Cited by 6 (0 self)
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The set B q p,r: = {⌊nq/p + r ⌋ : n ∈ Z} (with integers p,q,r) is a Beatty set with density p/q. We derive a formula for the Fourier transform ̂B q p∑ p,r(j): = e −2πij⌊nq/p+r⌋/q. n=1 A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m ≥ 3 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.
The Rat game and the Mouse game
, 2008
"... We define three new takeaway games, the Rat game, the Mouse game and the Fat Rat game. Three winning strategies are given for the Rat game and outlined for the Mouse and Fat Rat games. The efficiencies of the strategies are determined. Whereas the winning strategies of nontrivial takeaway games ar ..."
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Cited by 5 (3 self)
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We define three new takeaway games, the Rat game, the Mouse game and the Fat Rat game. Three winning strategies are given for the Rat game and outlined for the Mouse and Fat Rat games. The efficiencies of the strategies are determined. Whereas the winning strategies of nontrivial takeaway games are based on irrational numbers, our games are based on rational numbers. Another motivation stems from a problem in combinatorial number theory. 1 Description of the Game The Rat game is played on 3 piles of tokens by 2 players who play alternately. Positions in the game are denoted throughout in the form (x, y, z), with 0 ≤ x ≤ y ≤ z, and moves in the form (x, y, z) → (u, v, w), where of course also 0 ≤ u ≤ v ≤ w (see below). The player first unable to move — because the position is (0, 0, 0) — loses; the opponent wins. There are 3 types of moves: (I) Take any positive number of tokens from up to 2 piles.
Neglected possibilities of processing assertions and proofs mechanically: choice of problems and data, Universitylevel computer assisted instruction at
 Institute for Mathematical Studies in the Social Sciences
, 1981
"... FIFTY YEARS of research in logic have shown that mostofthe logicallynatural general problems cannot be solved mechanically. This research began in the thirties with the famous (recursive) incompleteness and undecidability results, and culminated in the seventies with large lower bounds for the numb ..."
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Cited by 5 (1 self)
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FIFTY YEARS of research in logic have shown that mostofthe logicallynatural general problems cannot be solved mechanically. This research began in the thirties with the famous (recursive) incompleteness and undecidability results, and culminated in the seventies with large lower bounds for the number of steps needed to execute earlier "positive " results. Thus, at least as long as the traditional parameters for classifying problems are used, for example, number of symbols (offormulae or derivations in any of the usual formal systems), essentially all known proof and decision procedures were shown to grow too fast for realistic use. These facts are by now weIl known, but not the obvious proviso that the negative results are tied to classes of problems chosen in traditional logical terms. The neglected possibilities mentioned in the title involve more sophisticated choices, of problems and data, so to speak, between the traditions of mathematical logic and of other systematic expositions, familiar for example from Bourbaki. The principal conclusions, supported in detail below, may be summarized