smooth integers
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2021 ◽  
Vol 219 ◽  
pp. 172-197
Author(s):  
Marzieh Mehdizadeh




2021 ◽  
pp. 1-1
Author(s):  
Vassil Dimitrov ◽  
Luigi Vigneri ◽  
Vidal Attias
Keyword(s):  


2016 ◽  
Vol 37 (2) ◽  
pp. 128-137 ◽  
Author(s):  
S. Ishmukhametov ◽  
F. Sharifullina
Keyword(s):  


2016 ◽  
Vol 152 (6) ◽  
pp. 1121-1158 ◽  
Author(s):  
Adam J. Harper

We investigate exponential sums over those numbers ${\leqslant}x$ all of whose prime factors are ${\leqslant}y$. We prove fairly good minor arc estimates, valid whenever $\log ^{3}x\leqslant y\leqslant x^{1/3}$. Then we prove sharp upper bounds for the $p$th moment of (possibly weighted) sums, for any real $p>2$ and $\log ^{C(p)}x\leqslant y\leqslant x$. Our proof develops an argument of Bourgain, showing that this can succeed without strong major arc information, and roughly speaking it would give sharp moment bounds and restriction estimates for any set sufficiently factorable relative to its density. By combining our bounds with major arc estimates of Drappeau, we obtain an asymptotic for the number of solutions of $a+b=c$ in $y$-smooth integers less than $x$ whenever $\log ^{C}x\leqslant y\leqslant x$. Previously this was only known assuming the generalised Riemann hypothesis. Combining them with transference machinery of Green, we prove Roth’s theorem for subsets of the $y$-smooth numbers whenever $\log ^{C}x\leqslant y\leqslant x$. This provides a deterministic set, of size ${\approx}x^{1-c}$, inside which Roth’s theorem holds.



2016 ◽  
Author(s):  
H. Ki ◽  
H. Maier ◽  
A. Sankaranarayanan
Keyword(s):  


2011 ◽  
Vol 97 (4) ◽  
pp. 319-324
Author(s):  
Filip Najman
Keyword(s):  


2011 ◽  
Vol 88 (11) ◽  
pp. 2222-2232 ◽  
Author(s):  
M. A. Mohamed ◽  
M. R. Md Said ◽  
K. A. Mohd Atan ◽  
Z. Ahmad Zulkarnain






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