Prescribing the binary digits of squarefree numbers and quadratic residues
Prescribing the binary digits of squarefree numbers and quadratic residues
复制标题
规定无平方数和二次余数的二进制数字
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
I. Shparlinski
中科院分区：
文献类型：
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作者：
R. Dietmann;Christian Elsholtz;I. Shparlinski
我们研究乘法定义的集合（例如无平方整数、二次非留数或原根）在以加法方式描述的集合（例如和集或希尔伯特立方）中的均匀分布。特别是，我们表明，如果固定给定二进制位长度的所有数字的数字中小于 $40\%$ 的任何比例，则剩余的集合仍然具有渐近预期的无平方整数数量。接下来，我们研究原根以大素数 $p$ 为模的分布，在原根集合中的希尔伯特立方体的最大维度上建立新的上限，从而改进了作者之前的结果。最后，我们研究有限域中的求和集，并渐进地找到此类求和集中二次留数和非留数的预期数量，前提是它们的基数足够大。这显着改进了 Dartyge、Mauduit 和 S\'ark\"ozy 最近的结果。我们的方法引入了几个新想法，结合了多种方法，例如指数和字符和的界限、数字几何和加法组合。
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 S\'ark\"ozy. 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.
DOI：
10.48550/arxiv.1207.0842
发表时间：
2012
期刊：
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影响因子：
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作者：
Dietmann R
通讯作者：
Dietmann R