Character sums over integers with restricted $g$-ary digits

Banks, William D.; Conflitti, Alessandro; Shparlinski, Igor E.

doi:10.1215/ijm/1258130986

Cited by 34 publications

(47 citation statements)

References 17 publications

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“…This follows from Lemma 2.3 (Weil's theorem) above by using an inequality of Vinogradov as presented by Niederreiter in [12] and Tietäväinen in [14] (see also [2], Lemma 2).…”

Section: Lemma 23 For Any Odd Prime P and Any Polynomialmentioning

confidence: 84%

On large families of subsets of the set of the integers not exceeding N

2008

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In an earlier paper Dartyge and Sárközy defined the measures of pseudorandomness of subsets of {1, 2, . . . , N}, and they presented several examples for subsets with strong pseudo-random properties. However, in applications one usually needs large families of subsets with strong pseudo-random properties. Here two constructions of this type are given. The notion of complexity of families of subsets of {1, . . . , N} is also introduced and studied.

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“…This follows from Lemma 2.3 (Weil's theorem) above by using an inequality of Vinogradov as presented by Niederreiter in [12] and Tietäväinen in [14] (see also [2], Lemma 2).…”

Section: Lemma 23 For Any Odd Prime P and Any Polynomialmentioning

confidence: 84%

On large families of subsets of the set of the integers not exceeding N

2008

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“…Hegyvári and Sárközy [23,Theorem 2] give the bound f (p) < 12p 1/4 for sufficiently large p, which has been improved to F (p) ≤ p 1/5+o (1) as p → ∞, by Dietmann, Elsholtz and Shparlinski [14,Theorem 1.3]. Here we improve this further to F (p) ≤ p 3/19+o(1) and recall that reducing the exponent below 1/8 immediately implies an improvement of the Burgess bound on the least primitive root (note that 3/19 − 1/8 = 0.0328 .…”

Section: 2mentioning

confidence: 99%

Prescribing the binary digits of squarefree numbers and quadratic residues

Dietmann

Elsholtz

Shparlinski

2017

Trans. Amer. Math. Soc.

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We study the equidistribution of multiplicatively defined sets, such as the squarefree integers, quadratic non-residues or primitive roots, in sets which are described in an additive way, such as sumsets or Hilbert cubes. In particular, we show that if one fixes any proportion less than 40% of the digits of all numbers of a given binary bit length, then the remaining set still has the asymptotically expected number of squarefree integers. Next, we investigate the distribution of primitive roots modulo a large prime p, establishing a new upper bound on the largest dimension of a Hilbert cube in the set of primitive roots, improving on a previous result of the authors. Finally, we study sumsets in finite fields and asymptotically find the expected number of quadratic residues and non-residues in such sumsets, given their cardinalities are big enough. This significantly improves on a recent result by Dartyge, Mauduit and Sárközy. Our approach introduces several new ideas, combining a variety of methods, such as bounds of exponential and character sums, geometry of numbers and additive combinatorics.

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“…The only property of Λ R1 (n), Λ R + 2 (m) that we will use from now on is that they are complex sequences bounded by O(log X) ℓ = (log X) Oη (1) . Explicitly, let α n = Λ R1 (n)n s2−s1 /(log X) ℓ+1 and β m = Λ R + 2 (m)m s2−s1 /(log X) ℓ+1 , and let γ a satisfy S A (a/X) = #Aγ a F X (a/X).…”

Section: Exceptional Minor Arcsmentioning

confidence: 99%

Primes with restricted digits

Maynard¹

2019

Invent. math.

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Let a 0 ∈ {0, . . . , 9}. We show there are infinitely many prime numbers which do not have the digit a 0 in their decimal expansion.The proof is an application of the Hardy-Littlewood circle method to a binary problem, and rests on obtaining suitable 'Type I' and 'Type II' arithmetic information for use in Harman's sieve to control the minor arcs. This is obtained by decorrelating Diophantine conditions which dictate when the Fourier transform of the primes is large from digital conditions which dictate when the Fourier transform of numbers with restricted digits is large. These estimates rely on a combination of the geometry of numbers, the large sieve and moment estimates obtained by comparison with a Markov process.

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Character sums over integers with restricted $g$-ary digits

Cited by 34 publications

References 17 publications

On large families of subsets of the set of the integers not exceeding N

On large families of subsets of the set of the integers not exceeding N

Prescribing the binary digits of squarefree numbers and quadratic residues

Primes with restricted digits

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