## Functiones et Approximatio Commentarii Mathematici

### On van der Corput property of shifted primes

Siniša Slijepčević

#### Abstract

We prove that the upper bound for the van der Corput property of the set of shifted primes is $O((\log n)^{-1+o(1)})$, giving an answer to a problem considered by Ruzsa and Montgomery for the set of shifted primes $p-1$. We construct normed non-negative valued cosine polynomials with the spectrum in the set $p-1$, $p\leq n$, and a small free coefficient $a_{0}=O((\log n)^{-1+o(1)})$. This implies the same bound for the intersective property of the set $p-1$, and also bounds for several properties related to uniform distribution of related sets.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 48, Number 1 (2013), 37-50.

Dates
First available in Project Euclid: 25 March 2013

https://projecteuclid.org/euclid.facm/1364222828

Digital Object Identifier
doi:10.7169/facm/2013.48.1.4

Mathematical Reviews number (MathSciNet)
MR3086959

Zentralblatt MATH identifier
1329.11076

#### Citation

Slijepčević, Siniša. On van der Corput property of shifted primes. Funct. Approx. Comment. Math. 48 (2013), no. 1, 37--50. doi:10.7169/facm/2013.48.1.4. https://projecteuclid.org/euclid.facm/1364222828

#### References

• J. Bourgain, Ruzsa's problem on sets of recurrence, Israel J. Math. 59 (1987), 151–166.
• T. Kamae and M. Mendès France, Van der Corput's difference theorem, Israel J. Math. 31 (1977), 335–342.
• J. Lucier, Difference sets and shifted primes, Acta Math. Hungar. 120 (2008), 79–102.
• H.L. Montgomery, Ten lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, AMS (1994), CMBS Regional Conference Series in Mathematics, 84.
• H.L. Montgomery and R.C. Vaughan, Multiplicative Number Theory, I. Classical Theory, Cambridge University Press (2007).
• I.Z. Ruzsa, Uniform distribution, positive trigonometric polynomials and difference sets, in Semin. on Number Theory. Univ. Bordeaux I, 1981-82. No. 18.
• I.Z. Ruzsa, Connections between the uniform distribution of a sequence and its differences, Topics in Classical Number Theory, Vol. I, II (Budapest, 1981), 1419–1443, Colloq. Math. Soc. Jànos Bolyai, 34, North-Holland, Amsterdam (1984).
• I.Z. Ruzsa, On measures of intersectivity, Acta Math. Hungar. 43 (1984), 335–340.
• I.Z. Ruzsa and T. Sanders, Difference sets and the primes, Acta Arith. 131 (2008), 281–301.
• A. Sárközy, On difference sets of integers III, Acta Math. Acad. Sci. Hungar. 31 (1978), 355–386.
• W.M. Schmidt, Small fractional parts of polynomials, Regional Conference Series No. 32, Amer. Mat. Soc., Providence (1977).
• S. Slijepčević, On van der Corput property of squares, Glas. Mat. Ser. III (2010), no. 2, 357–372.